Chapter 3: Machines
In the third century BCE, on the island of Sicily, the city of Syracuse had a king named Hiero II, and the king had a problem. A great ship had been built in his dockyard. It was heavy, it was loaded, and no gang of men pulling together could drag it down to the water.
Living in the same city was Archimedes, a mathematician who had been studying the behaviour of levers and pulleys. He had already made a remark that people remembered for centuries afterwards: give him a firm place to stand, he said, and he could move the Earth itself. The king, half amused, asked him to prove the claim on something rather smaller — the stranded ship.
Archimedes had ropes rigged around a system of pulleys and fastened to the hull. Then, according to the account written down by the historian Plutarch, he sat at a distance, took the free end of the rope, and drew it in steadily with his own hands. The ship slid forward as smoothly as if it were already afloat.
Nothing about Archimedes had changed. His muscles were no stronger that morning than the day before. What had changed was the arrangement of rope and wheels between his hands and the ship — and that arrangement is what this chapter is about.
Diagram
Fig. 3.0-1 — Archimedes Drawing A Laden Ship Towards The Sea Illustration of an ancient harbour scene: a large wooden ship resting on a slipway with ropes fastened to its bow; the ropes pass around a system of grooved wheels mounted in two blocks, one block tied to a fixed post on the shore and the other attached to the ship; Archimedes stands calmly at a distance holding the free end of the rope. Label the ship as the load, the free end of the rope as the effort, the fixed block, the movable block, and the several rope strands running between the two blocks.
3.1 Machines In Everyday Work
A nut that will not turn between your fingers comes loose easily when you fit a wrench over it. A heavy block that refuses to shift when you push it slides across the floor once you work a crowbar under its edge. In each case the job is the same, but the effort it demands is not.
A device that changes a job in this way is called a machine — an arrangement that lets us do a piece of work more conveniently than we could with our bare hands. The simplest of them have very few moving parts, and these are called simple machines: a wrench, a crowbar, a pulley, a spade, a pair of scissors.
Two words will be used constantly from here on. The load is the resistance the machine has to overcome, such as the weight of the block being lifted. The effort is the force you apply to the machine to overcome that load.
3.1.1 Functions And Uses Of Simple Machines
Machines are useful to us in four ways. Four panels of Fig. 3.1-1 show one everyday example of each.
Applying A Smaller Effort
The most familiar use of a machine is to overcome a large load with a small effort. A jack lifts a car when a person turns its handle; a bar lifts a heavy stone; a spade turns over packed soil; a pulley raises a heavy load.
In all of these the effort needed is much less than the load. A machine used in this way is called a force multiplier, because the force that reaches the load is larger than the force supplied by the hand.
Shifting The Point Of Application
Sometimes the difficulty is not the size of the force but where it has to be applied. The rear wheel of a bicycle is what has to be turned, yet nobody turns the wheel by hand.
The rider presses on the pedal, which turns a toothed wheel, and a chain carries the turning effect to the rear wheel. The effort is applied at the pedal, in easy reach of the feet, instead of at the wheel itself.
Changing The Direction Of Effort
A bucket lifted straight out of a well needs an effort pulling upwards, which is an awkward direction for the human body. Pass the rope over a single fixed pulley at the top of the well, and the bucket rises when the rope is pulled downwards.
The size of the effort has not been reduced here, only its direction has been turned to a convenient one. Pulling downwards is easier because a person can lean back and let their own weight help supply the effort.
Gaining Speed At The Load
The fourth use is the opposite of the first. When a pair of scissors cuts cloth, the handles move only a little while the blades sweep through a much longer distance, as shown in Fig. 3.1-2. The blade of a knife behaves the same way: a small movement of the handle carries the blade a long way.
Here the load moves further and faster than the effort does, and the machine is called a speed multiplier. The price of that gain is a heavier effort — in these examples the effort applied is larger than the load.
Diagram
Fig. 3.1-1 — Four Uses Of A Simple Machine Four small panels in a row, each labelled with load L and effort E and an arrow showing the direction of each force. Panel 1: a car jack raising the corner of a car, labelled ‘smaller effort than load’. Panel 2: a cyclist’s foot pressing a pedal, with the chain running to the toothed rear wheel, labelled ‘effort applied at a convenient point’. Panel 3: a bucket in a well with the rope passing over a fixed pulley and a person pulling the rope downwards, labelled ‘effort applied in a convenient direction’. Panel 4: a pair of scissors cutting cloth, labelled ‘load moves further than effort’.
Diagram
Fig. 3.1-2 — Blades Move Further Than The Handles A pair of scissors cutting a strip of cloth. Mark the rivet as the pivot. Draw a short curved arrow at the handles showing a small movement of the fingers, and a long curved arrow at the blades showing a much larger movement along the cloth. Label the handles as the point where effort is applied and the cloth as the load.
A machine can do one of these last two jobs, but never both at once. A force multiplier needs an effort smaller than the load; a speed multiplier needs an effort larger than the load. The same machine cannot need an effort that is both smaller and larger than the load, so no machine multiplies force and speed at the same time.
3.1.2 What A Machine Is
Everything in the last section can now be collected into a single statement. A machine either overcomes a large resistance with a small force, or moves the load faster than the effort moves — and in either case it lets us apply the force at a point and in a direction that suit us.
3.3 Principle Of A Machine
A machine does not create the work it delivers. Energy is supplied to it by the effort, and the machine spends that energy in overcoming the load. Everything a machine can and cannot do follows from this one fact.
3.3.1 The Effort Point And The Load Point
Every machine has two places worth watching. The effort point is the point at which energy is supplied to the machine by applying the effort. The load point is the point where energy is obtained from the machine by overcoming the load.
The energy supplied at the effort point is called the input energy, and it is simply the work done there:
\[\\text{input energy} = \\text{effort} \\times \\text{displacement of the point of application of effort}\]
The energy obtained at the load point is called the output energy, and it is the work obtained there:
\[\\text{output energy} = \\text{load} \\times \\text{displacement of the point of application of load}\]
Both are measured in joule (J), because both are work done. The two displacements are measured in metre (m) and the two forces in newton (N).
3.3.2 Ideal Machine And Actual Machine
Imagine a machine in which nothing at all is wasted — no rubbing at the joints, no weight to be lifted apart from the load, nothing that stretches or bends. Every joule handed to such a machine at the effort point comes out at the load point.
\[\\text{output energy} = \\text{input energy}\]
A machine of this kind is called an ideal machine: one in which there is no loss of energy in any manner, so that the work output equals the work input and the efficiency is \(100%\).
No real machine reaches that mark. In an actual machine the output energy is always less than the input energy, because some energy is always lost while the machine is working. The efficiency of an actual machine is therefore always less than \(100%\).
What is never possible is the other way round. The useful work done by a machine can never be greater than the work done on the machine, because that would mean energy appearing from nowhere, which would violate the principle of conservation of energy. No machine can ever have an efficiency greater than 1, that is, more than \(100%\).
3.3.3 Why Energy Is Lost In A Machine
The loss of energy in an actual machine is due to the following three reasons:
- The moving parts in it are neither weightless nor smooth (or frictionless).
- The string in it, if any, is not perfectly elastic.
- Its different parts are not perfectly rigid.
Of these, the most prominent loss of energy is in overcoming the force of friction between the moving parts of a machine. The energy so lost appears as heat, which is why the moving parts of a machine get heated.
Efficiency is the plainest way of stating how much survives. If a machine is \(80%\) efficient, it means that \(80%\) of the total energy supplied to the machine at the effort point is obtained as useful energy at the load point. The remaining \(20%\) of the energy supplied is lost in overcoming the force of friction and appears as heat in the different parts, which get heated up. Fig. 3.3-1 shows this sharing of the input energy.
Diagram
Fig. 3.3-1 — Sharing Of Input Energy In A Machine That Is 80 Per Cent Efficient A single horizontal bar representing 100 units of input energy entering at the effort point. Divide the bar into two portions: a long portion covering four fifths of its length, shaded and labelled ‘useful energy at the load point, 80%’, and a short portion covering the remaining fifth, labelled ‘energy spent against friction, 20%, appears as heat’. Draw an arrow into the left end of the bar labelled ‘input energy at the effort point’ and an arrow out of the shaded portion labelled ‘output energy at the load point’.
3.4 Relationship Between Efficiency, Mechanical Advantage And Velocity Ratio
Three quantities have now been defined, and they are not independent of one another. Mechanical advantage compares the two forces, velocity ratio compares the two distances, and efficiency compares the two amounts of work — and work is force times distance. A short derivation ties all three together.
Setting Up The Derivation
Suppose a machine overcomes a load \(L\) by the application of an effort \(E\), in time \(t\). Let the displacement of effort in that time be \(d\_E\) and the displacement of load be \(d\_L\). Then
\[W\_{\\text{input}} = E \\times d\_E \\qquad \\dots (\\text{i})\]
\[W\_{\\text{output}} = L \\times d\_L \\qquad \\dots (\\text{ii})\]
where \(E\) and \(L\) are in newton, \(d\_E\) and \(d\_L\) in metre, and both works in joule.
Working Through To The Relation
By definition, efficiency is the ratio of work output to work input. Substituting equations (i) and (ii),
\[\\eta = \\frac{W\_{\\text{output}}}{W\_{\\text{input}}} = \\frac{L \\times d\_L}{E \\times d\_E}\]
The right-hand side is a product of two forces and two distances, and it can be split so that the forces stay with the forces and the distances stay with the distances:
\[\\eta = \\frac{L}{E} \\times \\frac{d\_L}{d\_E}\]
The second fraction is upside down compared with the velocity ratio, which is \(d\_E/d\_L\). Writing it as a reciprocal instead,
\[\\eta = \\frac{L}{E} \\times \\frac{1}{d\_E/d\_L}\]
But \(\\dfrac{L}{E} = \\text{M.A.}\) and \(\\dfrac{d\_E}{d\_L} = \\text{V.R.}\), so
\[\\eta = \\frac{\\text{M.A.}}{\\text{V.R.}}\]
\[\\text{M.A.} = \\text{V.R.} \\times \\eta\]
Here M.A. is the mechanical advantage, V.R. the velocity ratio and \(\\eta\) the efficiency, all three being pure numbers with no unit. In words: the mechanical advantage of a machine is equal to the product of its efficiency and velocity ratio.
Reading The Relation For Ideal And Actual Machines
For an ideal machine, free from friction and from the weight of its own parts, the work output equals the work input, so the efficiency is 1 (or \(100%\)). Putting \(\\eta = 1\) in the relation gives \(\\text{M.A.} = \\text{V.R.}\) — the mechanical advantage of an ideal machine is numerically equal to its velocity ratio.
In actual practice some of the input energy is always lost against friction, so the efficiency is less than 1. For all practical machines, therefore, the mechanical advantage is always less than the velocity ratio, and the work output is always less than the work input.
This also settles a question left open in Section 3.2. Friction and the weight of the moving parts reduce the mechanical advantage, but the velocity ratio is decided by the arrangement of the parts alone and stays fixed for a given design. Since \(\\eta = \\text{M.A.}/\\text{V.R.}\), a falling mechanical advantage divided by an unchanged velocity ratio means a falling efficiency.
3.5 Levers
Levers are the simplest kind of machines used in our daily life. A stick under a stone, a spoon under the lid of a tin, a pair of tongs at the fire — each is a bar that turns, and each makes a job easier by turning.
3.5.1 The Lever And Its Fulcrum
The axis about which the lever turns passes through a point of the lever which is called the fulcrum. It is generally marked by the letter F. This point, or the axis through it, does not move but remains fixed when the lever is in use.
Fig. 3.5-1 shows a lever, a straight rod AB, with the fulcrum at F. An effort \(E\) applied at a point A of the lever overcomes a load \(L\) at the point B. Two distances measured from the fulcrum decide everything that follows. The distance FA, from the fulcrum to the point at which the effort is applied, is called the effort arm; the distance FB, from the fulcrum to the point at which the load acts, is called the load arm.
For an ideal lever it is assumed that the rod is weightless and that there is no friction at the fulcrum. Only under these two assumptions do the simple results of this section hold exactly.
Diagram
Fig. 3.5-1 — The Two Arms Of A Lever A straight horizontal rod AB resting on a triangular fulcrum marked F, placed between A and B and nearer to B. At end A draw a downward arrow labelled effort E; at end B draw a downward arrow labelled load L. Mark the distance from F to A with a double-headed arrow labelled ‘effort arm FA’ and the distance from F to B with a double-headed arrow labelled ‘load arm FB’. Add a curved arrow near B showing the clockwise turning caused by the load, and a curved arrow near A showing the anticlockwise turning caused by the effort.
3.5.2 Principle Of A Lever
A lever works by turning, and the turning effect of a force about a point is its moment of a force — the force multiplied by the perpendicular distance of its line of action from that point, measured in newton metre (N m). The load, pressing down at B, tries to turn the rod one way about F; the effort, pressing down at A, tries to turn it the other way. When the lever is balanced, the two turning effects cancel.
In Fig. 3.5-1 the moment of the load \(L\) about the fulcrum F is clockwise, while the moment of the effort \(E\) about the fulcrum F is anticlockwise. Thus,
\[\\text{clockwise moment of load about the fulcrum} = \\text{anticlockwise moment of effort about the fulcrum}\]
\[\\text{Load} \\times \\text{load arm} = \\text{Effort} \\times \\text{effort arm}\]
\[L \\times \\text{FB} = E \\times \\text{FA}\]
where \(L\) and \(E\) are in newton and the two arms are in metre. The two arms must always be measured in the same unit; if both are in centimetre, the relation still holds, because the units cancel when the two sides are compared.
3.5.3 The Law Of Levers
Rearranging the balance of moments so that the two forces stand together on one side and the two arms on the other,
\[\\frac{L}{E} = \\frac{\\text{FA}}{\\text{FB}}\]
But \(L/E\) is exactly the mechanical advantage of the machine. Hence
\[\\text{M.A.} = \\frac{\\text{effort arm FA}}{\\text{load arm FB}}\]
This relation is known as the law of levers. It holds for an ideal lever, that is, one whose bar is weightless and whose fulcrum is frictionless.
The law makes the behaviour of any lever readable at a glance, simply by comparing the two arms.
- If effort arm \(=\) load arm, then \(\\text{M.A.} = 1\).
- If effort arm \(>\) load arm, then \(\\text{M.A.} > 1\).
- If effort arm \(<\) load arm, then \(\\text{M.A.} < 1\).
Obviously, then, the mechanical advantage of a lever can be increased either by increasing its effort arm, or by decreasing its load arm — and the load arm is shortened by shifting the fulcrum towards the load.
3.6 Kinds Of Levers
Every lever has the same three things on it: a fulcrum, a point where the load acts and a point where the effort is applied. What differs from one lever to another is the order in which these three lie along the bar.
Depending upon the relative positions of effort, load and fulcrum, there are the following three types of levers: Class I levers, Class II levers and Class III levers. Fig. 3.6-1 shows the three arrangements side by side.
Diagram
Fig. 3.6-1 — The Three Classes Of Levers Three horizontal bars drawn one below the other, each labelled with its class. (a) Class I: a bar with the fulcrum F marked by a triangle in the middle, the effort E at the left end shown by a downward arrow and the load L at the right end shown by a downward arrow; mark the effort arm from F to E and the load arm from F to L. (b) Class II: a bar with the fulcrum F as a triangle at the left end, the load L in between shown by a downward arrow, and the effort E at the right end shown by an upward arrow; mark the load arm from F to L and the longer effort arm from F to E. (c) Class III: a bar with the fulcrum F as a triangle at the left end, the effort E in between shown by an upward arrow, and the load L at the right end shown by a downward arrow; mark the shorter effort arm from F to E and the longer load arm from F to L.
3.6.1 Class I Levers
In this type of levers, the fulcrum F is in between the effort E and the load L, as shown in Fig. 3.6-1 (a). Note that the fulcrum F need not be at the mid-point between the load L and the effort E.
The load and effort are on either side of the fulcrum, acting in the same direction while producing rotation in opposite sense. Both may press downwards, but because they press on opposite sides of the fulcrum, one turns the bar clockwise and the other anticlockwise.
Since the fulcrum can sit anywhere along the bar, either arm may be the longer one. For class I levers, therefore, the mechanical advantage and velocity ratio can have any value — either greater than 1, equal to 1, or less than 1.
A seesaw, a pair of scissors, a crowbar, the handle of a water pump, a claw hammer, a pair of pliers, the beam of a physical balance, a spade used for turning soil, a spoon used to open the lid of a tin can, a catapult and the nodding of the human head are a few examples of class I levers. Some of these are shown in Fig. 3.6-2.
Diagram
Fig. 3.6-2 — Some Class I Levers Six small labelled sketches, each marked with the fulcrum F, an arrow for the load L and an arrow for the effort E. (i) A pair of pliers gripping an object, with the rivet as F. (ii) A seesaw with a child at each end and the central pivot as F. (iii) A crowbar with its tip under a block and a stone under the bar as F. (iv) A claw hammer pulling a nail out of a plank, with the head resting on the plank as F. (v) A pair of shears cutting a metal sheet, with the rivet as F, drawn with long handles and short blades. (vi) A spoon prising up the lid of a tin can, with the rim of the can as F.
When The Effort Arm Is The Longer One
When the effort arm is longer than the load arm, the mechanical advantage and the velocity ratio of an ideal class I lever are greater than 1. Such a lever serves as a force multiplier: it enables us to overcome a large resistive force with a small effort.
Shears used for cutting thin metal sheets have much longer handles as compared with their blades. Similarly a crowbar, a claw hammer, a pair of pliers and a spoon used to open the lid of a container all have long handles, that is, long effort arms.
When The Two Arms Are Equal
If an ideal class I lever has effort arm and load arm of equal lengths, then both its mechanical advantage and velocity ratio are equal to 1. A physical balance, with both arms equal in length, is exactly of this kind. It gains neither force nor speed; what it does is compare two weights fairly.
When The Load Arm Is The Longer One
When an ideal lever of class I has its effort arm shorter than its load arm, both its mechanical advantage and velocity ratio are less than 1. Such levers are used to obtain a gain in speed, because a velocity ratio less than 1 implies that \(d\_L > d\_E\), that is, the displacement of the load is more than the displacement of the effort.
The pair of scissors met in Section 3.1 is exactly this arrangement: its blades are longer than its handles, so that the blades move a long way along the cloth when the handles are moved a little.
A long handle oar used for rowing a boat by a single person acts as a lever of class I, as shown in Fig. 3.6-3. The point at the edge of the boat at which the handle rests acts as the fulcrum F. The boatman applies the effort E at its one end, keeping the effort arm shorter than the load arm, so as to give a large movement to the blade of the oar. The blade pushes water, which is the load L, back through a longer distance — a gain in speed — and the force of reaction exerted by water on the boat moves the boat forward. Its mechanical advantage is less than 1.
Diagram
Fig. 3.6-3 — Rowing A Boat With A Long Handle Oar A single rower seated in a boat on water, holding the handle of a long oar. Mark the point where the oar rests on the edge of the boat as the fulcrum F, the rower’s hands at the handle end as the effort E with an arrow, and the blade of the oar in the water as the load L with an arrow. Show that the distance from F to the handle is shorter than the distance from F to the blade.
3.6.2 Class II Levers
In this type of levers, the fulcrum F and the effort E are at the two ends of the lever, and the load L is somewhere in between the effort E and the fulcrum F, as shown in Fig. 3.6-1 (b). The load and effort are on the same side of the fulcrum but acting in opposite directions.
Because the load always sits between the fulcrum and the effort, the effort is always further from the fulcrum than the load is. Here the effort arm is always longer than the load arm. Therefore, from the law of levers, the mechanical advantage is always greater than 1, and since in an ideal machine the mechanical advantage is equal to the velocity ratio, the velocity ratio is also greater than 1.
In other words, a class II lever always acts as a force multiplier: less effort is needed to overcome a large load. For example, in a nut cracker a hard nut is broken by applying a small effort.
A nut cracker, a bottle opener, a wheel barrow, a lemon crusher, a paper cutter, a mango cutter, a bar used to lift a load, a door, and the raising of the weight of the human body on toes are examples of class II levers. Some of these are shown in Fig. 3.6-4.
Diagram
Fig. 3.6-4 — Some Class II Levers Four small labelled sketches, each marked with the fulcrum F, the load L and the effort E with arrows. (i) A nut cracker with a nut held between its arms, the hinge as F, the nut as L and the hand at the open ends as E. (ii) A bottle opener lifting the cap of a bottle, with the far lip of the opener resting on the cap rim as F. (iii) A wheel barrow carrying a load, with the centre of the wheel as F, the load in the tray as L and the hands at the ends of the handles as E, the effort arrow pointing upwards. (iv) A straight bar prising up a heavy block, with one end of the bar on the ground as F and the block resting on the bar near that end as L.
3.6.3 Class III Levers
In this type of levers, the fulcrum F and the load L are at the two ends of the lever, and the effort E is somewhere in between the fulcrum F and the load L, as shown in Fig. 3.6-1 (c). Effort and load are on the same side of the fulcrum but acting in opposite directions.
Now it is the effort that lies between the other two, so the effort arm is always smaller than the load arm. Therefore, from the law of levers, the mechanical advantage is less than 1; and since for an ideal lever the mechanical advantage is equal to the velocity ratio, the velocity ratio is also less than 1 for these levers.
With levers of class III we do not get a gain in force, but we get a gain in speed: a larger displacement of load is obtained by a smaller displacement of effort. For example, the blade of a knife moves a long way when its handle is displaced a little.
Sugar tongs, the forearm used for lifting a load (the action of the biceps muscle), fire tongs, a foot treadle, a knife, a spade used to lift coal or soil, and a fishing rod are examples of class III levers. Some of these are shown in Fig. 3.6-5.
Diagram
Fig. 3.6-5 — Some Class III Levers Four small labelled sketches, each marked with the fulcrum F, the load L and the effort E with arrows. (i) A pair of sugar tongs lifting a sugar cube, with the closed bend as F, the fingers pressing partway along the arms as E and the cube at the tips as L. (ii) A foot treadle, with the hinged end as F, the foot pressing part way along as E and the connecting rod at the far end as L. (iii) A knife slicing, with the far end of the handle as F, the hand as E and the blade in contact with the material as L. (iv) A spade lifting a load of soil, with the upper hand at the top of the shaft as F, the lower hand partway down as E and the soil on the blade as L.
The short handle oar used in a boat race by boatmen acts as a lever of class III. In Fig. 3.6-6, each boatman sits near the edge of the boat, one after the other, all facing towards the front. Each boatman holds the top end of the oar stationary with one hand, and this end acts as the fulcrum F. He applies the effort E on the handle at some distance from the fulcrum with his other hand. The blade of each oar pushes the water, which acts as load L at the other end of the oar, and the reaction force pushes the boat forward. The oar is then raised out of the water and the process is repeated. Since the effort arm is shorter than the load arm, each oar provides a gain in speed, and all the boatmen row the boat in unison so as to obtain more gain in speed and win the boat race.
Diagram
Fig. 3.6-6 — Rowing With Short Handle Oars In A Boat Race A long racing boat seen from above and slightly to one side, with several rowers seated one behind the other near the edges on both sides, all facing the front of the boat; an arrow at the front labelled ‘direction of motion of boat’. On one oar, mark the top end held steady by one hand as the fulcrum F, the other hand on the handle as the effort E with an arrow, and the blade in the water as the load L with an arrow. Show the distance from F to E as shorter than the distance from F to L.
Table 3.2 gathers the three classes together for comparison.
3.7 Examples Of Each Class Of Lever In The Human Body
In a human body we can find examples of all the three classes of levers. The bones act as the rigid bars and the joints act as the fulcrums. The muscles exert the force, that is, the effort, by contraction — a muscle shortens and pulls the bone it is attached to.
Nodding Of The Head
Fig. 3.7-1 shows the action of nodding of the head. In this action the spine acts as the fulcrum F, the load L is at the front part of the skull, while the effort E is applied at its rear part by the muscles at the back of the neck. The fulcrum lies between the load and the effort, so this is an example of a class I lever.
Diagram
Fig. 3.7-1 — Nodding Of The Head As A Class I Lever Side view of a human head and the top of the neck. Mark the joint where the skull rests on the spine as the fulcrum F, an arrow at the front part of the skull pointing downwards labelled load L, and an arrow at the rear part of the skull pointing downwards labelled effort E applied by the neck muscles.
Raising The Body On The Toes
Fig. 3.7-2 shows how the weight of the body is raised on the toes. The fulcrum F is at the toes at one end, the load L, which is the weight of the body, is in the middle, and the effort E is applied at the other end by the muscles of the calf, which pull the heel upwards. The load lies between the fulcrum and the effort, so this is an example of a class II lever.
Diagram
Fig. 3.7-2 — Raising The Weight Of The Body On The Toes As A Class II Lever Side view of a foot and lower leg raised on tiptoe. Mark the toes in contact with the ground as the fulcrum F, a downward arrow at the ankle joint labelled load L, being the weight of the body, and an upward arrow at the heel labelled effort E applied by the calf muscle through the tendon.
Raising A Load On The Palm
Fig. 3.7-3 shows the action of the biceps. The elbow joint acts as the fulcrum F at one end, the biceps exerts the effort E in the middle, and the load L on the palm is at the other end. The effort lies between the fulcrum and the load, so this is an example of a class III lever.
Diagram
Fig. 3.7-3 — Raising A Load Kept On The Palm As A Class III Lever Side view of a bent forearm with a weight resting on the open palm. Mark the elbow joint as the fulcrum F, an upward arrow a short distance along the forearm where the biceps is attached to the bone, labelled effort E, and a downward arrow at the palm labelled load L.
Because the effort arm here is much the shorter of the two, the muscle must pull with a force far larger than the weight being held — as the next example shows.
3.8 The Pulley
A lever gains its advantage from the length of a bar. The next machine gains its advantage from a wheel and a rope, and it is the machine Archimedes reached for when the ship would not move.
3.8.1 Construction Of A Simple Pulley
Fig. 3.8-1 shows a simple pulley. It is a metallic or wooden disc with a grooved rim. A string or rope is passed around the groove at the rim. The disc rotates about an axle passing through its centre, and the axle is fixed rigidly to a frame by means of nails.
Diagram
Fig. 3.8-1 — A Simple Pulley Two views side by side. (a) Front view: a circular disc with a groove running round its rim, an axle through its centre, and a frame straddling the disc and fixed to it at the axle; label the disc, the grooved rim, the axle, the frame and the nails holding the frame. (b) Side view of the same pulley, showing the frame on either side of the disc, the axle passing through the centre, and a string lying in the groove of the rim; label the disc, grooved rim, axle, frame, nail and string.
The groove is what keeps the rope from slipping sideways off the wheel. When the rope is pulled, the pulley turns with it, and it is the force due to friction between the string and the surface of the rim of the pulley which rotates the pulley when the string is pulled down.
3.8.2 The Two Ways Of Using A Pulley
A single pulley can be used in two ways. It may be used as a fixed pulley, by keeping its axis of rotation stationary, that is, by keeping its frame fixed in position. Or it may be used as a movable pulley, by changing its axis of rotation, that is, by keeping the whole frame movable so that the pulley itself travels along with the load.
The same wheel, the same rope and the same groove behave quite differently in the two arrangements. The next two sections take each in turn.
3.9 Single Fixed Pulley
Fig. 3.9-1 shows a single fixed pulley in which the axle is fixed to a rigid support, and an inextensible strong string of negligible mass passes around the grooved rim of the pulley. One end of the string is connected to the load \(L\), while the effort \(E\) is applied at the other, free, end of the string. Both the load \(L\) and the effort \(E\) act downwards.
On pulling the string downwards by the effort, the pulley rotates and the load moves up. The tension throughout the string is the same, written \(T\), and it acts upwards on both sides of the pulley. The tension has to be the same everywhere along the string, for if it were not, the string itself would move even when the pulley was not rotating. This type of pulley is used for lifting a small load, such as a basket.
Diagram
Fig. 3.9-1 — A Single Fixed Pulley A pulley hanging from a rigid support at the top, drawn with its frame and axle; label the rigid support, the frame, the axle and the string. The string passes over the grooved rim. On the left side the string goes down to a load hanging at its end, with a downward arrow labelled load L and an upward arrow along the string labelled tension T. On the right side a person pulls the free end downwards, with a downward arrow labelled effort E and an upward arrow along that side of the string also labelled tension T.
3.9.1 Mechanical Advantage, Velocity Ratio And Efficiency Of A Single Fixed Pulley
To find the ideal values, neglect two things: the mass of the string, and the friction at the axle, that is, in the pulley bearings. In the balanced position of the load, the string on the load side holds the load up, so
\[L = T \\qquad \\dots(\\text{i})\]
and the hand at the free end has only to match the tension on that side, so, when the pulley is not rotating,
\[E = T \\qquad \\dots(\\text{ii})\]
Both \(L\) and \(E\) are equal to the same tension \(T\), and therefore, from equations (i) and (ii),
\[E = L\]
Thus, in the case of a single fixed pulley, in an ideal case, the effort needed to lift a load is equal to the load itself. If the load is \(10\\ \\text{kgf}\), the effort needed is also \(10\\ \\text{kgf}\).
The mechanical advantage follows at once:
\[\\text{M.A.} = \\frac{\\text{load } L}{\\text{effort } E} = \\frac{T}{T} = 1\]
Thus, in this arrangement there is no gain in mechanical advantage.
Now look at the distances. If the point of application of the effort \(E\) moves a distance \(d\) downwards, the load \(L\) also moves the same distance \(d\) upwards, because the string is inextensible and whatever length leaves one side arrives at the other. So if \(d\_E = d\), then \(d\_L = d\), and
\[\\text{V.R.} = \\frac{d\_E}{d\_L} = \\frac{d}{d} = 1\]
\[\\eta = \\frac{\\text{M.A.}}{\\text{V.R.}} = \\frac{1}{1} = 1 \\text{ or } 100%\]
In actual practice, however, there is always some friction at the axle or in the pulley bearings, so the effort needed is a little more than the load to be lifted. The mechanical advantage is then less than 1, while the velocity ratio remains 1, so the efficiency of an actual single fixed pulley is always less than \(100%\).
3.9.2 Why A Fixed Pulley Is Used
A fair question now arises. If there is no gain in mechanical advantage and no gain in speed either, why is a fixed pulley used at all?
A fixed pulley is used only to change the direction of the effort to be applied. With its use, the effort can be applied in a more convenient direction.
It is difficult to apply an effort upwards to lift a load up directly, but it becomes easier with the help of a fixed pulley, because the effort can now be applied in the downward direction to raise the load up. Further, to apply the effort downwards, one can conveniently make use of one’s own weight also for the effort.
3.10 Single Movable Pulley
Fig. 3.10-1 shows a single movable pulley, suspended by passing an inextensible string of negligible mass around its grooved rim. The load \(L\) to be raised is suspended from its axle. One end of the string is tied to a hook H fixed at a rigid support, and the effort \(E\) is applied at its free end. The tension \(T\) in the string acts on both sides of the pulley.
Diagram
Fig. 3.10-1 — A Single Movable Pulley A rigid support at the top with a hook labelled H. One end of a string is tied to the hook and hangs down, passes under and around the grooved rim of a pulley, and rises again on the other side to a free end. Label the frame and axle of the pulley. Hang a load from the axle of the pulley with a downward arrow labelled load L. Draw an upward arrow along each of the two string segments, both labelled tension T. At the free end draw an upward arrow labelled effort E, and mark that this end is equal to T.
3.10.1 Mechanical Advantage, Velocity Ratio And Efficiency Of A Single Movable Pulley
For the ideal values, neglect two things again: the friction in the pulley bearings or at the axle, and the weight of the pulley and string.
The load hangs from the axle, and the pulley itself is held up by two segments of string, each pulling upwards with tension \(T\). The load is therefore balanced by both of them together:
\[L = T + T = 2T \\qquad \\dots(\\text{i})\]
The effort has only to equal the tension at the free end of the string, so
\[E = T \\qquad \\dots(\\text{ii})\]
From equations (i) and (ii),
\[E = \\frac{L}{2}\]
Thus, using a single movable pulley, a load can be lifted by applying an effort equal to half the load in the ideal situation. A single movable pulley therefore acts as a force multiplier. Its mechanical advantage is
\[\\text{M.A.} = \\frac{\\text{load } L}{\\text{effort } E} = \\frac{2T}{T} = 2\]
Now for the distances. When the free end of the string is pulled up by the effort through a distance \(d\), the load is raised up through a distance \(d/2\). The reason is that the segment of string on both sides of the pulley must shorten by \(d/2\) each to give up a total length \(d\) at the free end. So if \(d\_E = d\), then \(d\_L = d/2\), and
\[\\text{V.R.} = \\frac{\\text{distance moved by the effort } d\_E}{\\text{distance moved by the load } d\_L} = \\frac{d}{d/2} = 2\]
\[\\eta = \\frac{\\text{M.A.}}{\\text{V.R.}} = \\frac{2}{2} = 1 \\text{ or } 100%\]
For an actual single movable pulley, due to the friction in the pulley bearings or at the axle, and the weight of the pulley and string, the effort needed to lift up a load \(L\) will be more than \(L/2\), so the mechanical advantage will be less than 2. Since the velocity ratio remains 2, the efficiency will always be less than 1 or \(100%\).
3.10.2 Changing The Direction Of Effort With A Movable Pulley
While lifting a load with a single movable pulley, the effort is to be applied in an upward direction. However, it is inconvenient to apply an effort in an upward direction.
Therefore a movable pulley A is used along with a fixed pulley B to change the direction of the effort, as shown in Fig. 3.10-2. The load is attached to the axle of the movable pulley A, and the effort is applied in the downward direction at the free end of the string passing over the fixed pulley B. One can also use one’s own weight as effort, which is quite convenient in such a case.
The fixed pulley here does no lifting of its own. The mechanical advantage and the velocity ratio of this arrangement remain the same as that of a single movable pulley — in the ideal case, \(\\text{M.A.} = 2\) and \(\\text{V.R.} = 2\).
Diagram
Fig. 3.10-2 — A Movable Pulley Used With A Fixed Pulley A rigid support at the top carrying two attachments. On the left, one end of a string is tied to the support; the string passes down and under the grooved rim of a movable pulley A, then rises and passes over a fixed pulley B mounted on the same support to the right, and finally hangs down as a free end. Label pulley A as the movable pulley and pulley B as the fixed pulley. Hang the load from the axle of pulley A with a downward arrow labelled load L. Mark the tension T along each string segment, and draw a downward arrow at the free end labelled effort E.
3.10.3 Single Fixed Pulley Compared With Single Movable Pulley
Table 3.3 sets the two arrangements side by side.
3.11 Combination Of Pulleys
When a heavy load is to be lifted or shifted from one place to another, we require a pulley system of mechanical advantage much more than 2, so a single movable pulley is not enough. A combination of several pulleys with one or more strong thick strings or ropes is then used.
The combination can be made in two ways:
- Using one fixed pulley and several movable pulleys attached to the same rigid support.
- Using several pulleys in two blocks, of which the lower block is movable and the upper block is fixed. This is known as the block and tackle system.
3.11.1 One Fixed Pulley With Several Movable Pulleys
Fig. 3.11-1 shows a system of three movable pulleys A, B and C used with a fixed pulley D. Each movable pulley is suspended with a separate string around its rim. Because the strings are separate, the tension is the same within any one string but different in different strings, being maximum in the string of the pulley to which the load is attached.
Diagram
Fig. 3.11-1 — System Of One Fixed Pulley And Three Movable Pulleys A rigid support at the top. Three movable pulleys A, B and C are drawn one above the other, with A lowest, and a fixed pulley D is mounted on the support at the right. The load hangs from the axle of the lowest pulley A, marked with a downward arrow labelled load L. A separate string passes round pulley A, with one end tied to the rigid support and the other end tied to the axle of pulley B; label both segments of this string with tension T1. A second string passes round pulley B, one end tied to the support and the other tied to the axle of pulley C; label its segments T2. A third string passes round pulley C, one end tied to the support and the other passing over the fixed pulley D to hang down as the free end; label its segments T3 and mark the free end with a downward arrow labelled effort E.
Mechanical Advantage Of The System
In equilibrium, the effort simply balances the tension in the last string, which passes over the fixed pulley D:
\[E = T\_3 \\qquad \\dots(\\text{i})\]
The two segments of the string passing over the pulley A support the load \(L\), therefore the tension \(T\_1\) in this string is given by \(2T\_1 = L\), or
\[T\_1 = \\frac{L}{2} \\qquad \\dots(\\text{ii})\]
Similarly, the two segments of the string passing over the pulley B support the tension \(T\_1\), so \(2T\_2 = T\_1\), and using equation (ii),
\[T\_2 = \\frac{T\_1}{2} = \\frac{L}{2^2} \\qquad \\dots(\\text{iii})\]
Similarly, the tension \(T\_3\) in the string passing over the pulley C is given by \(2T\_3 = T\_2\), and using equation (iii),
\[T\_3 = \\frac{T\_2}{2} = \\frac{L}{2^3} \\qquad \\dots(\\text{iv})\]
From equation (iv), \(L = 2^3 \\times T\_3\), and comparing this with equation (i),
\[E = \\frac{L}{2^3}\]
Hence the mechanical advantage is
\[\\text{M.A.} = \\frac{\\text{load } L}{\\text{effort } E} = \\frac{2^3 \\times T\_3}{T\_3} = 2^3\]
FC 3.2 shows the same result as a chain: each pulley in turn halves the force that has to be carried upwards.
Flowchart
FC 3.2 — How The Tension Falls At Each Movable Pulley Load L hanging from pulley A → string round A carries tension T₁ = L/2 → this string pulls the axle of pulley B → string round B carries tension T₂ = L/4 → this string pulls the axle of pulley C → string round C carries tension T₃ = L/8 → string passes over fixed pulley D → effort E = L/8 applied downwards
In general, if there are \(n\) movable pulleys with one fixed pulley, the mechanical advantage is then
\[\\text{M.A.} = 2^n\]
where \(n\) is the number of movable pulleys, and the mechanical advantage carries no unit.
Velocity Ratio Of The System
As one end of each string passing over a movable pulley is fixed, the other end of the string moves up twice the distance moved by the axle of the movable pulley.
If the load \(L\) attached to the pulley A moves up by a distance \(x\), that is \(d\_L = x\), then the string connected to the axle of pulley B moves up by a distance \(2 \\times x = 2x\). The string connected to the axle of pulley C then moves up by a distance \(2 \\times 2x = 2^2 x\), and the end of the string passing over the fixed pulley D moves up by a distance \(2 \\times 2^2 x = 2^3 x\). That is, the effort \(E\) moves by a distance \(2^3 x\), so \(d\_E = 2^3 x\).
\[\\text{V.R.} = \\frac{\\text{distance moved by the effort } d\_E}{\\text{distance moved by the load } d\_L} = \\frac{2^3 x}{x} = 2^3\]
In general, if there are \(n\) movable pulleys connected to a fixed pulley, the velocity ratio is
\[\\text{V.R.} = 2^n\]
\[\\eta = \\frac{\\text{M.A.}}{\\text{V.R.}} = \\frac{2^n}{2^n} = 1 \\text{ or } 100%\]
The efficiency of this arrangement is \(100%\) only in an ideal situation. In actual practice both the weight of the movable pulleys and string, and the friction between the bearings of the pulleys, reduce the mechanical advantage, and thus the efficiency becomes less than \(100%\).
3.11.2 The Block And Tackle System
In this system of pulleys, two blocks of pulleys are used. One block, the upper one, having several pulleys, is attached to a rigid support and is therefore fixed; the other block, the lower one, also having several pulleys, is movable.
For the convenience of applying effort in the downward direction, the number of pulleys in the movable lower block is kept either equal to, or one less than, the number of pulleys in the fixed upper block. A strong inextensible string or rope of negligible mass passes around all the pulleys.
One end of the string is attached to the hook of the lower block if the number of pulleys in the upper block is more than that in the lower block, or it is attached to the hook of the upper block if the number of pulleys is equal in both the blocks, so as to apply the effort in the downward direction. Fig. 3.11-2 shows a block and tackle system of 5 pulleys, in which one end of the string is attached to the hook of the lower block, while Fig. 3.11-3 shows a block and tackle system of 4 pulleys in which one end of the string is attached to the hook of the upper block.
The load \(L\) is attached to the movable lower block and the effort \(E\) is applied at the free end of the string. Since it is one single rope running round every pulley, the tension along the entire length of the string is the same and is denoted as \(T\).
Diagram
Fig. 3.11-2 — Block And Tackle System Of Five Pulleys An upper block of three pulleys mounted one below the other and fixed to a rigid support at the top, and a lower block of two pulleys mounted one below the other. A single continuous rope has one end tied to the hook of the lower block, then passes alternately around each pulley of the upper and lower blocks in turn, and finally comes down as a free end. Label each of the five rope segments running between the two blocks with tension T, draw a downward arrow at the free end labelled effort E, and hang the load from the hook at the bottom of the lower block with a downward arrow labelled load L.
Diagram
Fig. 3.11-3 — Block And Tackle System Of Four Pulleys An upper block of two pulleys fixed to a rigid support at the top, and a lower block of two pulleys. A single continuous rope has one end tied to the hook of the upper block, then passes alternately around each pulley of the lower and upper blocks in turn, and finally comes down as a free end. Label each of the four rope segments supporting the lower block with tension T, draw a downward arrow at the free end labelled effort E, and hang the load from the hook at the bottom of the lower block with a downward arrow labelled load L.
Mechanical Advantage Of A Block And Tackle
In Fig. 3.11-2, the tension in the five segments of the string supports the load \(L\). Therefore \(L = 5T\) and \(E = T\), so
\[E = \\frac{L}{5}, \\qquad \\text{M.A.} = \\frac{L}{E} = \\frac{5T}{T} = 5\]
Similarly, in Fig. 3.11-3, the tension in the four segments of the string supports the load \(L\). Therefore \(L = 4T\) and \(E = T\), so
\[E = \\frac{L}{4}, \\qquad \\text{M.A.} = \\frac{L}{E} = \\frac{4T}{T} = 4\]
In general, if the total number of pulleys used in both the blocks is \(n\), and the effort is applied in the downward direction, then the tension in the \(n\) segments of the string supports the load. Therefore
\[L = nT \\quad \\text{and} \\quad E = T, \\quad \\text{so} \\quad E = \\frac{L}{n}\]
\[E = \\frac{L \\ (\\text{load})}{n \\ (\\text{number of pulleys})}\]
Thus, in a block and tackle system, the effort gets multiplied \(n\) times, where \(n\) is the total number of pulleys in the system. It therefore acts as a force multiplier. Hence
\[\\text{M.A.} = \\frac{\\text{load } L}{\\text{effort } E} = \\frac{nT}{T} = n = \\text{total number of pulleys in both the blocks}\]
Velocity Ratio And Efficiency
In a system of \(n\) pulleys, if the load moves up through a distance \(d\), each segment of the string supporting the load is loosened by a length \(d\), so the effort end moves through a distance \(nd\). That is, if \(d\_L = d\), then \(d\_E = nd\), and
\[\\text{V.R.} = \\frac{d\_E}{d\_L} = \\frac{nd}{d} = n\]
Thus, the velocity ratio is always equal to the number of strands of the tackle, or the number of sections of the string, supporting the load.
\[\\eta = \\frac{\\text{M.A.}}{\\text{V.R.}} = \\frac{n}{n} = 1 \\text{ or } 100%\]
Efficiency is \(100%\) only in an ideal situation. But in actual practice there is always some friction in the bearings of the pulleys, and also the weight of the string and the weight of the lower block with pulleys is not negligible, so the mechanical advantage decreases and hence the efficiency decreases.
No Gain In Energy
A pulley system is only a force multiplier and, just like other machines, there is no gain in energy. This can be seen as follows. The work done by the effort, that is the input energy, is
\[W\_{\\text{input}} = E \\times nd = nEd\]
The work done on the load, that is the output energy, is
\[W\_{\\text{output}} = L \\times d = nE \\times d = nEd\]
using \(L = nE\), which follows from \(\\text{M.A.} = L/E = n\). The two are equal, so there is no gain in energy — only a redistribution of force and distance.
3.11.3 Effect Of The Weight Of The Pulleys
So far the lower block has been treated as weightless. In a real tackle it is not, and the effort has to support that weight as well as the load.
Consider a system of \(n\) pulleys. Let \(w\) be the total weight of the lower block along with the pulleys in it, measured in newton. In the balanced position, the effort equals the tension in the rope, and the \(n\) strands together must support both the load and the lower block:
\[E = T \\quad \\text{and} \\quad L + w = nT\]
\[L = nT - w = nE - w\]
Hence the mechanical advantage is
\[\\text{M.A.} = \\frac{L}{E} = \\frac{nE - w}{E} = n - \\frac{w}{E}\]
where \(n\) is the total number of pulleys, \(w\) the weight of the lower block in newton, and \(E\) the effort in newton. Since \(w/E\) is a positive quantity, the mechanical advantage is always less than the ideal value \(n\).
The velocity ratio does not change, because it depends only on how many strands support the load, not on how heavy they are. It remains
\[\\text{V.R.} = n\]
\[\\eta = \\frac{\\text{M.A.}}{\\text{V.R.}} = \\frac{n - \\dfrac{w}{E}}{n} = 1 - \\frac{w}{nE}\]
Thus, the efficiency is reduced due to the weight of the lower block of pulleys. The more the weight of the lower block, the less is the efficiency.
For greater efficiency, therefore, the pulleys in the lower block should be as light as possible, and the friction in the bearings of the pulleys should be minimised by the use of lubricants.
3.12 Summary
A machine is a device by which we can either obtain a gain in speed or overcome a large resistive force (or load) at some point, by applying a small force (or effort) at a convenient point and in a desired direction.
Machines are useful to us in four ways: in lifting a heavy load by applying less effort, in changing the point of application of effort to a convenient point, in changing the direction of effort to a convenient direction, and in obtaining a gain in speed.
The load \(L\) is the resistive or opposing force to be overcome by a machine, and the effort \(E\) is the force applied on the machine to overcome the load. Both are measured in newton (N).
Mechanical advantage is the ratio of load to effort: \[\\text{M.A.} = \\frac{L}{E}\] where \(L\) is the load in N and \(E\) the effort in N. Being a ratio of two forces, it has no unit.
A machine with \(\\text{M.A.} > 1\) acts as a force multiplier; with \(\\text{M.A.} < 1\) it gives a gain in speed; with \(\\text{M.A.} = 1\) it only changes the direction of the effort. A machine cannot be a force multiplier and a speed multiplier at the same time.
Velocity ratio is the ratio of the velocity of effort to the velocity of load, and equally the ratio of their displacements in the same time: \[\\text{V.R.} = \\frac{V\_E}{V\_L} = \\frac{d\_E}{d\_L}\] where \(d\_E\) is the distance moved by the effort in m and \(d\_L\) the distance moved by the load in m. It has no unit.
Work input is the work done on the machine by the effort, and work output is the work done by the machine on the load: \[W\_{\\text{input}} = E \\times d\_E, \\qquad W\_{\\text{output}} = L \\times d\_L\] both measured in joule (J).
Efficiency is the ratio of work output to work input: \[\\eta = \\frac{W\_{\\text{output}}}{W\_{\\text{input}}} \\times 100%\] where both works are in J. It has no unit.
The point at which energy is supplied to a machine is the effort point, and the point where energy is obtained by overcoming the load is the load point. Input energy \(=\) effort \(\\times\) displacement of the point of application of effort; output energy \(=\) load \(\\times\) displacement of the point of application of load.
For an ideal machine, output energy \(=\) input energy, and its efficiency is \(100%\). In an actual machine the output energy is always less than the input energy, so its efficiency is always less than \(100%\).
No machine can have an efficiency greater than \(100%\), as that would violate the principle of conservation of energy.
Energy is lost in an actual machine because its moving parts are neither weightless nor frictionless, its string, if any, is not perfectly elastic, and its parts are not perfectly rigid. The most prominent loss is against friction, and the energy so lost appears as heat.
The three quantities are related by \[\\eta = \\frac{\\text{M.A.}}{\\text{V.R.}}, \\qquad \\text{M.A.} = \\text{V.R.} \\times \\eta\] For an ideal machine \(\\eta = 1\), so \(\\text{M.A.} = \\text{V.R.}\); for all practical machines \(\\text{M.A.} < \\text{V.R.}\)
Friction and the weight of the moving parts reduce the mechanical advantage, but the velocity ratio of a given design does not change, so the efficiency falls.
A lever is a rigid, straight or bent bar which is capable of turning about a fixed axis. The fixed point about which it turns is the fulcrum F. The distance of the effort from the fulcrum is the effort arm, and the distance of the load from the fulcrum is the load arm.
A lever works on the principle of moments: at the equilibrium position of the lever, the moment of the load about the fulcrum is equal to the moment of the effort about the fulcrum, and the two moments are in opposite directions. \[L \\times \\text{load arm} = E \\times \\text{effort arm}\]
The law of levers: the mechanical advantage of a lever is equal to the ratio of the length of its effort arm to the length of its load arm. \[\\text{M.A.} = \\frac{\\text{effort arm}}{\\text{load arm}}\] This holds for an ideal lever, one whose bar is weightless and whose fulcrum is frictionless.
The mechanical advantage of a lever is increased by increasing its effort arm or by decreasing its load arm, that is, by shifting the fulcrum towards the load.
In a class I lever the fulcrum is between the effort and the load, and the mechanical advantage and velocity ratio may be more than 1, equal to 1 or less than 1. Examples: a seesaw, a crowbar, a claw hammer, a pair of pliers, the beam of a physical balance.
In a class II lever the load is between the fulcrum and the effort, so the effort arm is always longer and the mechanical advantage and velocity ratio are always more than 1. Such a lever is always a force multiplier. Examples: a nut cracker, a bottle opener, a wheel barrow.
In a class III lever the effort is between the fulcrum and the load, so the effort arm is always shorter and the mechanical advantage and velocity ratio are always less than 1. Such a lever gives a gain in speed. Examples: sugar tongs, fire tongs, a knife, a fishing rod.
All three classes occur in the human body: the nodding of the head is a class I lever, raising the weight of the body on the toes is a class II lever, and raising a load on the palm using the biceps is a class III lever.
A pulley is a disc with a grooved rim, rotating about an axle fixed in a frame. A fixed pulley has its axis of rotation stationary in position; a movable pulley has its axis of rotation movable.
For an ideal single fixed pulley, \(E = L\), so \(\\text{M.A.} = 1\), \(\\text{V.R.} = 1\) and \(\\eta = 100%\). It gives no gain in force and no gain in speed, and is used only to change the direction of the effort to a convenient one.
For an ideal single movable pulley, \(E = L/2\), so \(\\text{M.A.} = 2\), \(\\text{V.R.} = 2\) and \(\\eta = 100%\). It acts as a force multiplier. A fixed pulley is often used along with it to apply the effort downwards, without altering the mechanical advantage or the velocity ratio.
For a combination of \(n\) movable pulleys with one fixed pulley, each movable pulley carrying its own string, \[\\text{M.A.} = 2^n, \\qquad \\text{V.R.} = 2^n, \\qquad \\eta = 100% \\text{ (ideal)}\]
In a block and tackle system of \(n\) pulleys in two blocks, with a single rope and the effort applied downwards, \[E = \\frac{L}{n}, \\qquad \\text{M.A.} = n, \\qquad \\text{V.R.} = n, \\qquad \\eta = 100% \\text{ (ideal)}\] The velocity ratio is always equal to the number of strands of the tackle supporting the load.
A pulley system gives no gain in energy: the work done by the effort equals the work done on the load in the ideal case.
If the lower block of a block and tackle has a total weight \(w\) in N, then \[\\text{M.A.} = n - \\frac{w}{E}, \\qquad \\text{V.R.} = n, \\qquad \\eta = 1 - \\frac{w}{nE}\] so the mechanical advantage and the efficiency both fall. For greater efficiency, the pulleys of the lower block should be as light as possible and the bearings should be lubricated.
Flowchart
FC 3.3 — Analysing Any Machine Identify the load L and the effort E → find the mechanical advantage, M.A. = L/E → find the velocity ratio from the geometry of the machine: the ratio of the arms for a lever, or the number of supporting strands for a pulley system → find the efficiency, η = M.A. / V.R. → compare with the ideal machine, for which M.A. = V.R. and η = 100%
3.13 Key Terms
- Actual machine — a real machine, in which some energy is always lost, so that its work output is less than its work input and its efficiency is less than 100 per cent.
- Block and tackle system — a pulley system using several pulleys in two blocks, the upper block fixed and the lower block movable, with one continuous rope passing round all of them.
- Efficiency — the ratio of the work output of a machine to its work input, usually expressed as a percentage; it has no unit.
- Effort — the force applied on a machine to overcome the load; SI unit newton (N).
- Effort arm — the distance of the point of application of the effort from the fulcrum of a lever; SI unit metre (m).
- Effort point — the point at which energy is supplied to a machine by applying the effort.
- Fixed pulley — a pulley whose axis of rotation is stationary in position.
- Force multiplier — a machine in which the effort needed is less than the load, so that its mechanical advantage is greater than 1.
- Fulcrum — the fixed point, or the axis through it, about which a lever turns; it does not move while the lever is in use.
- Ideal lever — a lever assumed to have a weightless bar and no friction at its fulcrum.
- Ideal machine — a machine in which there is no loss of energy in any manner, so that its work output equals its work input and its efficiency is 100 per cent.
- Input energy — the energy supplied to a machine at the effort point, equal to the effort multiplied by the displacement of its point of application; SI unit joule (J).
- Law of levers — the statement that the mechanical advantage of a lever equals the ratio of the length of its effort arm to the length of its load arm.
- Lever — a rigid, straight or bent bar which is capable of turning about a fixed axis.
- Load — the resistive or opposing force to be overcome by a machine; SI unit newton (N).
- Load arm — the distance of the point at which the load acts from the fulcrum of a lever; SI unit metre (m).
- Load point — the point where energy is obtained from a machine by overcoming the load.
- Machine — a device by which we can either obtain a gain in speed or overcome a large resistive force at some point, by applying a small force at a convenient point and in a desired direction.
- Mechanical advantage — the ratio of the load to the effort for a machine; it has no unit.
- Moment of a force — the turning effect of a force about a point, equal to the force multiplied by the perpendicular distance of its line of action from that point; SI unit newton metre (N m).
- Movable pulley — a pulley whose axis of rotation is movable, so that the pulley itself travels along with the load.
- Output energy — the energy obtained from a machine at the load point, equal to the load multiplied by the displacement of its point of application; SI unit joule (J).
- Pulley — a metallic or wooden disc with a grooved rim, which rotates about an axle fixed in a frame, with a string passing round the groove.
- Simple machine — a machine with very few moving parts, such as a lever, a pulley, an inclined plane or a wedge.
- Speed multiplier — a machine in which the load moves further and faster than the effort, so that its mechanical advantage is less than 1.
- Velocity ratio — the ratio of the velocity of the effort to the velocity of the load, equal to the ratio of their displacements in the same time; it has no unit.
- Work input — the work done on a machine by the effort; SI unit joule (J).
- Work output — the work done by a machine on the load; SI unit joule (J).
3.14 Points To Ponder
A machine never delivers more energy than it receives, yet we say a crowbar makes the work easier. If the work itself is not reduced, what exactly has been made easier — and is ‘easier’ a physical quantity at all?
Most levers in the human body are class III, which cost force rather than save it. What advantage might a limb gain from an arrangement whose mechanical advantage is always less than 1, and in what kinds of movement would that advantage matter most?
The velocity ratio of a machine is fixed by its geometry alone. Is there any way to change the velocity ratio of a given lever without changing the length of its bar? What does your answer say about why velocity ratio is unaffected by friction?
Suppose friction could be removed completely from a block and tackle. Would the load then stay where it was when the free end of the rope was released? What does your answer suggest about a use of friction that is not a nuisance?
An ideal single fixed pulley gains neither force nor speed, and its efficiency is 100 per cent. Is a device that gains nothing still worth calling a machine? Frame your reasoning in terms of the definition given in this chapter.
Efficiency can never exceed 100 per cent, but could a real machine ever reach exactly 100 per cent? List everything that would have to be true of its parts, its string and its bearings, and decide whether such a machine could exist.
3.15 Objective Questions
- (MCQ) A machine whose mechanical advantage is less than 1 is used to obtain: (1 mark)
- a gain in force
- a gain in speed
- a gain in energy
- a gain in efficiency
- (MCQ) For an ideal machine, the correct statement is: (1 mark)
- work output is greater than work input
- work output is less than work input
- work output is equal to work input
- work output is independent of work input
- (MCQ) The velocity ratio of a machine depends upon: (1 mark)
- the friction at its moving parts
- the arrangement of its parts
- the size of the load placed on it
- the weight of its moving parts
- (MCQ) In which class of lever is the effort arm always shorter than the load arm? (1 mark)
- Class I
- Class II
- Class III
- Class II and class III both
- (MCQ) The ideal mechanical advantage of a single fixed pulley is: (1 mark)
- \(\\tfrac{1}{2}\)
- 1
- 2
- 4
- (MCQ) A block and tackle system has 3 pulleys in the upper block and 2 in the lower block, and the effort is applied downwards. Its velocity ratio is: (1 mark)
- 2
- 3
- 5
- 6
- (MCQ) If \(n\) movable pulleys are used with one fixed pulley, each movable pulley carrying its own string, the ideal mechanical advantage is: (1 mark)
- \(n\)
- \(2n\)
- \(n^2\)
- \(2^n\)
- (MCQ) Which of these quantities of a machine has a unit? (1 mark)
- Mechanical advantage
- Velocity ratio
- Efficiency
- Work output
- (Assertion–Reason) Assertion (A): The efficiency of an actual machine is always less than 100 per cent. Reason (R): Some of the input energy of a machine is always spent in overcoming friction and appears as heat. (1 mark)
- Both A and R are true and R is the correct explanation of A
- Both A and R are true but R is not the correct explanation of A
- A is true but R is false
- A is false but R is true
- (Assertion–Reason) Assertion (A): A single fixed pulley is used even though it gives no gain in force. Reason (R): The velocity ratio of a single fixed pulley is 2. (1 mark)
- Both A and R are true and R is the correct explanation of A
- Both A and R are true but R is not the correct explanation of A
- A is true but R is false
- A is false but R is true
- (Match the following) Match each machine in column A with the correct description in column B. (4 marks)
| Column A | Column B |
|---|---|
| a. Wheel barrow | i. Class I lever |
| b. Sugar tongs | ii. Class II lever |
| c. Physical balance | iii. Class III lever |
| d. Single movable pulley | iv. Ideal mechanical advantage 2 |
(Fill in the blank) The mechanical advantage of a machine is equal to the product of its velocity ratio and its __________. (1 mark)
(Fill in the blank) In a class II lever, the __________ lies between the fulcrum and the effort. (1 mark)
(Fill in the blank) The velocity ratio of a block and tackle system is equal to the number of __________ of the tackle supporting the load. (1 mark)
(True or false) A machine can act as a force multiplier and a speed multiplier at the same time. (1 mark)
(True or false) The velocity ratio of a machine falls when friction increases at its moving parts. (1 mark)
(True or false) The mechanical advantage of a lever increases when its fulcrum is shifted towards the load. (1 mark)
(Name the following) The point of a lever about which it turns and which itself does not move. (1 mark)
(Name the following) The class of lever formed by the human forearm when a load is raised on the palm. (1 mark)
(Name the following) The pulley system in which two blocks of pulleys are used with a single continuous rope. (1 mark)
(Give one word or phrase for) The work done by a machine on the load. (1 mark)
(Give one word or phrase for) A machine in which there is no loss of energy in any manner. (1 mark)
(State the SI unit of) The load acting on a machine. (1 mark)
(State the SI unit of) The input energy supplied at the effort point of a machine. (1 mark)
(State the SI unit of) The moment of a force about the fulcrum of a lever. (1 mark)
(Odd one out) Nut cracker, bottle opener, fishing rod, wheel barrow. Give a reason for your choice. (1 mark)
(Odd one out) Mechanical advantage, velocity ratio, work input, efficiency. Give a reason for your choice. (1 mark)
3.16 Check Your Concepts
Define the mechanical advantage of a machine and state its unit. (2 marks)
State the four ways in which a machine is useful to us. (2 marks)
Give a reason why the mechanical advantage of an actual machine is always less than its velocity ratio. (2 marks)
State the principle of a machine, and say what is meant by calling a machine ideal. (2 marks)
Why is a single fixed pulley used at all, when it gives neither a gain in force nor a gain in speed? (2 marks)
Differentiate between a force multiplier and a speed multiplier, stating clearly the basis of difference. (3 marks)
State the principle of moments as applied to a lever, giving the condition under which it holds. (3 marks)
Give two reasons why the efficiency of an actual single movable pulley is less than 100 per cent. (3 marks)
Explain why the mechanical advantage of a class II lever is always more than 1, while that of a class III lever is always less than 1. (3 marks)
Why does the mechanical advantage of a machine fall when friction increases, while its velocity ratio stays the same? Explain what happens to the efficiency. (3 marks)
Explain why scissors meant for cutting cloth have blades longer than their handles, while shears meant for cutting metal sheets have short blades and long handles. (4 marks)
Class III levers demand an effort larger than the load. Why, then, are they used at all, and why are so many of them found in the human body? Support your answer with one example. (4 marks)
Derive the relation between the mechanical advantage, the velocity ratio and the efficiency of a machine. (5 marks)
Describe a block and tackle system of 4 pulleys. State where one end of the rope is attached and why, and derive expressions for its mechanical advantage and velocity ratio in the ideal case. (5 marks)
What would happen to the efficiency of a block and tackle system if the lower block were made heavier? Explain your answer using the relation for mechanical advantage, and state two ways of improving the efficiency of such a system. (5 marks)
3.17 Diagrams, Graphs & Constructions
Draw a labelled diagram of a class I lever in which the mechanical advantage is greater than 1. Mark the fulcrum F, the load L, the effort E, the effort arm and the load arm. (3 marks)
Fig. 3.17-1 shows a uniform bar resting on a fulcrum. Identify the class of lever, and copy the figure marking the effort arm and the load arm. State whether the mechanical advantage is more than, equal to, or less than 1, and give a reason. (3 marks)
Diagram
Fig. 3.17-1 — A Bar In Use As A Lever A straight horizontal bar. A triangular fulcrum marked F is placed at the left end of the bar. A downward arrow marked L acts on the bar at a point one quarter of the way along from F. An upward arrow marked E acts at the far right end of the bar. Do not mark any arms or distances.
Draw a neat labelled diagram of a single movable pulley used along with a single fixed pulley to raise a load. Mark the load L, the effort E and the tension T in each segment of the string, and state the ideal mechanical advantage of the arrangement. (4 marks)
Fig. 3.17-2 shows a block and tackle system with the effort applied downwards. Count the number of strands supporting the load, and hence state the velocity ratio of the system. Copy the figure and mark the tension in each strand and the directions of the load and the effort. (4 marks)
Diagram
Fig. 3.17-2 — A Block And Tackle System An upper block containing two pulleys is fixed to a rigid support at the top, and a lower block containing two pulleys hangs below it. A single rope has one end tied to the hook of the upper block and passes alternately round the pulleys of the lower and upper blocks, coming down at the right as a free end. Show the rope path clearly, but do not label the load, the effort or the tensions.
Draw a labelled diagram to show a combination of three movable pulleys used with one fixed pulley. Mark the directions of the load, the effort and the tension in each string, and state the ideal mechanical advantage of the combination. (4 marks)
Draw a labelled sketch of a class III lever found in the human body. Mark the fulcrum, the load and the effort, and name the joint and the muscle involved. (3 marks)
3.18 Practice With Numericals
An ideal lever has an effort arm of \(75\\ \\text{cm}\) and a load arm of \(15\\ \\text{cm}\). Find its mechanical advantage, and the effort needed to raise a load of \(450\\ \\text{N}\). (3 marks)
A crowbar \(150\\ \\text{cm}\) long is pivoted at a point \(25\\ \\text{cm}\) from the load. Find its mechanical advantage, and the effort needed to lift a load of \(900\\ \\text{N}\). (3 marks)
In a machine, an effort of \(40\\ \\text{N}\) moves through \(3.0\\ \\text{m}\) while a load of \(90\\ \\text{N}\) is raised through \(1.0\\ \\text{m}\). Calculate the work input, the work output and the efficiency of the machine. (3 marks)
A machine has a velocity ratio of 6 and an efficiency of 50 per cent. Find its mechanical advantage, and the effort needed to raise a load of \(720\\ \\text{N}\). (3 marks)
A single movable pulley raises a load of \(300\\ \\text{N}\) with an effort of \(200\\ \\text{N}\). Find its mechanical advantage, its velocity ratio and its efficiency. (3 marks)
A pair of fire tongs has arms \(24\\ \\text{cm}\) long, and the effort is applied at a distance of \(8\\ \\text{cm}\) from the fulcrum. Find the class of the lever, its mechanical advantage, and the effort needed to lift a piece of coal of weight \(6\\ \\text{N}\). (3 marks)
A uniform seesaw \(4.0\\ \\text{m}\) long is pivoted at its centre. A boy of weight \(400\\ \\text{N}\) sits at a distance of \(1.5\\ \\text{m}\) from the centre. At what distance from the centre, and on which side, must a girl of weight \(300\\ \\text{N}\) sit so as to balance him? (4 marks)
A lever is used to raise a load of \(30\\ \\text{kgf}\). Its effort arm is \(90\\ \\text{cm}\) and its load arm is \(18\\ \\text{cm}\). Taking \(g = 10\\ \\text{m s}^{-2}\), find the load in newton, the mechanical advantage of the lever, and the effort needed in newton. (4 marks)
Three movable pulleys are used with one fixed pulley, each movable pulley carrying its own string, to raise a load of \(1600\\ \\text{N}\). Assuming the system to be ideal, find its mechanical advantage, its velocity ratio and the effort needed. If the load is raised through \(0.25\\ \\text{m}\), find the distance moved by the effort. (4 marks)
A block and tackle system of 5 pulleys, with the effort applied downwards, raises a load of \(1500\\ \\text{N}\) through \(0.40\\ \\text{m}\) using an effort of \(400\\ \\text{N}\). Find the velocity ratio, the mechanical advantage, the efficiency, the distance moved by the effort, and the work done by the effort. (5 marks)
A block and tackle system has 4 pulleys, and the total weight of its lower block along with its pulleys is \(40\\ \\text{N}\). An effort of \(250\\ \\text{N}\) is applied downwards. Neglecting friction, find the load raised, the mechanical advantage and the efficiency of the system. (5 marks)