ICSE Class 10 Physics Chapter 03 Machines

Chapter 3: Machines

Syllabus

Machines as force multipliers: load, effort, mechanical advantage, velocity ratio and efficiency; simple treatment of levers; pulley systems showing the utility of each type of machine.

Scope of syllabus: Functions and uses of simple machines. Terms: effort E, load L, mechanical advantage (M.A.), velocity ratio (V.R.), efficiency (η), relation between η and M.A., V.R. (derivation included); for all practical machines η - 1; M.A. - V.R.

Lever: Principle, First, second and third class of levers; examples; M.A. and V.R. in each case. Examples of each of these classes of levers as also found in the human body.

Pulley system: Single fixed, single movable, block and tackle; M.A., V.R. and η in each case.

A - Machines, Technical Terms and Levers

3.1 Machines

It is our common experience that it is easier to open a nut by the use of a wrench than to open it by hand. It is difficult to pull up a bucket of water directly from a well, but it becomes much easier to pull it up with the use of a pulley. It is difficult to shift a heavy block by pushing it, but it becomes easier to shift it by using a crow bar. We can find many such examples in our daily life where the use of a machine (such as the wrench, pulley, crow bar, etc.) makes the job easier.

Functions and uses of simple machines:

The various functions of machines are useful to us in the following four ways:

(1) In lifting a heavy load by applying the less effort, i.e., as a force multiplier.

Examples: A jack is used to lift a car, a bar is used to lift a heavy stone, a spade is used to turn the soil, pulleys are used to lift a heavy load. In all these examples, the effort is much less than the load, so the machine acts as a force multiplier.

(2) In changing the point of application of effort to a convenient point.

Example: The rear wheel of a cycle is rotated with the help of a chain joined to a toothed wheel by applying the effort on the pedal attached with it. Thus the point of application of effort is changed from rear wheel to the pedal.

(3) In changing the direction of effort to a convenient direction.

Example: If a bucket, full of water, is lifted up from a well without the use of a pulley, effort has to be applied upwards. By using a single fixed pulley, it becomes possible to lift the bucket from the well by applying the effort in the downward direction instead of applying it upwards so that the person lifting it up may also use his own weight as effort.

(4) For obtaining a gain in speed (i.e., a greater movement of load by a smaller movement of effort).

Examples: (i) When a pair of scissors is used to cut the cloth, its blades move longer on cloth, while its handles move a little. (ii) The blade of a knife moves longer

by a small displacement of its handle. Here the effort is more than the load.

Hence we define a machine as below:

A machine is a device by which we can either 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 or by which we can obtain a gain in speed.

Note: A machine can not be used as a force multiplier as well as a speed multiplier simultaneously.

3.2 Technical Terms Related to a Machine

(1) Load

The resistive or opposing force to be overcome by a machine is called the load (L).

(2) Effort

The force applied on the machine to overcome the load is called the effort (E).

(3) Mechanical Advantage (M.A.)

The ratio of the load to the effort is called the mechanical advantage of the machine, i.e.,

Mechanical advantage (M.A.) = Load (L) / Effort (E)

Unit: Since mechanical advantage is the ratio of two similar quantities, so it has no unit.

While using a machine to overcome a certain load, if the effort needed is less than the load, the machine has the mechanical advantage greater than 1, while if it needs an effort greater than the load, it has the mechanical advantage less than 1. A machine has the mechanical advantage equal to 1, if the effort needed is equal to the load. A machine having the mechanical advantage greater than 1, acts as a force multiplier, while the machine having the mechanical advantage less than 1, gives the gain in speed. The machine having the mechanical advantage equal to 1, is generally used to change the direction of effort as there is no gain in force or speed.

(4) Velocity Ratio (V.R.)

The ratio of the velocity of effort to the velocity of load is called the velocity ratio of machine, i.e.,

Velocity ratio (V.R.) = Velocity of effort (V subscript E) / Velocity of load (V subscript L)

If d subscript L and d subscript E are the distances moved in same time t by the load and the effort respectively, then

Velocity of load (V subscript L) = d subscript L / t

Velocity of effort (V subscript E) = d subscript E / t

Velocity ratio (V.R.) = V subscript E / V subscript L = (d subscript E / t) / (d subscript L / t)

or

V.R. = d subscript E / d subscript L

Thus, the velocity ratio is also defined as the ratio of the displacement of effort to the displacement of load.

A machine in which the displacement of load is more than the displacement of effort, will have the velocity ratio less than 1 and such a machine gives the gain in speed because load is moving at a faster rate. On the other hand, if the velocity ratio of machine is more than 1, i.e., the displacement of load is less than the displacement of effort, the machine acts as a force multiplier. The velocity ratio of a machine is 1 if the displacement of load is equal to the displacement of effort. Such a machine generally changes the direction of effort.

Unit: Since the velocity ratio is also the ratio of two similar quantities, so it has no unit just like M.A.

(5) Work Input

The work done on the machine by the effort, is called the work input (W subscript input), i.e.,

Work input = work done by the effort.

(6) Work Output

The work done by the machine on the load, is called the work output (W subscript output), i.e.,

Work output = work done on the load.

(7) Efficiency (η)

Efficiency of a machine is the ratio of the work done on load by the machine to the work done on the machine by the effort.

In other words, efficiency is the ratio of the work output to the work input. It is denoted by the symbol η (eta). Thus

Efficiency η = Work output (W subscript output) / Work input (W subscript input)

But efficiency is usually expressed as a percentage, so we may write

Efficiency η = Work output (W subscript output) / Work input (W subscript input) × 100 %

Unit: It has no unit since it is also the ratio of two similar quantities (i.e., work).

Note: The presence of friction and weight of the moving parts of a machine of a given design, have no effect on its velocity ratio, but decreases both, its mechanical advantage and efficiency.

For an ideal machine,

Output energy = Input energy

The useful work done by a machine (i.e., output energy) can never be greater than the work done on the machine (i.e., input energy), otherwise it will violate the principle of conservation of energy, therefore no machine can ever have efficiency greater than 1 (i.e., more than 100%).

Ideal machine: An ideal machine is that in which there is no loss of energy in any manner. Here the work output is equal to the work input. i.e., the efficiency of an ideal machine is 100%.

Actual machine: In an actual machine, the output energy is always less than the input energy indicating that there is some loss of energy during its operation. The loss in energy is mainly due to the following three reasons:

(i) the moving parts in it are neither weightless nor smooth (or frictionless),

(ii) the string in it (if any) is not perfectly elastic, and

(iii) its different parts are not perfectly rigid.

Note: The energy lost in overcoming the force of friction between the moving parts of a machine, is the most common type of loss of energy in it.

If a machine is 80% efficient, it implies 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 etc. and it appears as heat energy in its different parts as a result they get heated up.

3.3 Principle of a Machine

When energy is supplied to a machine by applying the effort, it overcomes the load by doing some useful work on it.

The point at which the energy is supplied to a machine by applying the effort is called the effort point and the point where the energy is obtained by overcoming the load, is called the load point.

Input energy = work done at the effort point = effort × displacement of the point of application of effort.

Output energy = work obtained at the load point = load × displacement of the point of application of load.

3.4 Relationship Between Efficiency (η), Mechanical Advantage (M.A.) and Velocity Ratio (V.R.)

Suppose a machine overcomes a load L by the application of an effort E, in time t. Let the displacement of effort be d subscript E and the displacement of load be d subscript L.

Work input = effort × displacement of effort = E × d subscript E

Work output = load × displacement of load = L × d subscript L

By definition,

Efficiency η = work output / work input

From eqns. (i) and (ii),

η = (L × d subscript L) / (E × d subscript E) = (L / E) × (d subscript L / d subscript E) = (L / E) × 1 / (d subscript E / d subscript L)

But L / E = M.A. and d subscript E / d subscript L = V.R.

Efficiency η = M.A. / V.R.

or

M.A. = V.R. × η

Thus, the mechanical advantage of a machine is equal to the product of its efficiency and velocity ratio.

Note: For an ideal machine (free from friction, etc.), work output is equal to the work input, so the efficiency is equal to 1 (or 100%) and the mechanical advantage is numerically equal to the velocity ratio. In actual practice, the mechanical advantage for all practical machines is always less than its velocity ratio (i.e., M.A. - V.R.) or the output work is always less than the input work, so the efficiency is less than 1 (i.e., η - 1) due to some loss of input energy against friction, etc.

3.5 Levers

Levers are the simplest kind of machines used in our daily life.

A lever is a rigid, straight (or bent) bar which is capable of turning about a fixed axis.

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 does not move, but remains fixed when the lever is in use.

Principle of a Lever (M.A. of a Lever)

Fig. 3.1 shows a lever (or 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. From the fulcrum F, the distance FA to the point A at which the effort is applied, is called the effort arm and the distance FB of point B at which the load acts, is called the load arm. For an ideal lever, it is assumed that the rod is weightless and there is no friction at the fulcrum.

A lever works on the principle of moments according to which in the equilibrium position of lever, moment of load about the fulcrum must be equal to the moment of effort about the fulcrum and the two moments must always be in opposite direction. In Fig. 3.1, moment of load L about the fulcrum F is clockwise, while the moment of effort E about the fulcrum F is anticlockwise. Thus

Clockwise moment of load about the fulcrum = Anticlockwise moment of effort about the fulcrum.

i.e., Load × load arm = Effort × effort arm

or L × FB = E × FA

or

L / E = FA / FB

But from eqn. (3.1) L / E = M.A.

M.A. = Effort arm FA / Load arm FB

This relation is known as the law of levers. Thus

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.

From eqn. (3.10), it is clear that

(1) if effort arm = load arm, M.A. = 1,

(2) if effort arm - load arm, M.A. - 1, and

(3) if effort arm - load arm, M.A. - 1.

Obviously the mechanical advantage of a lever can be increased either by increasing its effort arm or by decreasing its load arm.

3.6 Kinds of Levers

Depending upon the relative positions of the effort, load and fulcrum, there are following three types of levers: (1) Class I levers, (2) Class II levers, and (3) Class III levers.

(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. 3.1 (a). Note that the fulcrum F need not be at the mid-point between the load L and the effort E, but both the load and effort are in same direction.

Examples: A seesaw, a pair of scissors, crowbar, handle of water pump, claw hammer, pair of pliers, beam of a physical balance, spade used for turning the soil, spoon used to open the lid of a tin can, catapult and nodding of the human head are, the few examples of Class I levers. Some of these are shown in Fig. 3.2.

For class I levers, the mechanical advantage and velocity ratio can have any value either greater than 1, equal to 1 or less than 1.

When the effort arm is longer than the load arm, the mechanical advantage and the velocity ratio of the class I lever are greater than 1. Such a lever serves as a force multiplier, i.e., it enables us to overcome a large resistive force (load) by a small effort. For example, shears (used for cutting the thin metal sheets) have much longer handles as compared to its blades. Similarly a crowbar, claw hammer, pliers and spoon (used to open the lid of a container) have long handles (or long effort arm).

If a class I lever has effort arm and load arm of equal lengths, then both its mechanical advantage and velocity ratio are equal to 1. For example, a physical balance with both arms equal in length has mechanical advantage and velocity ratio equal to 1.

When a lever of class I has the effort arm shorter than the load arm, both its mechanical advantage and velocity ratio are less than 1. Such levers are used to obtain the gain in speed because the velocity ratio less than 1 implies that d subscript L - d subscript E i.e., the displacement of load is more as compared to the displacement of effort. For example, a pair of scissors, whose blades are longer than its handles, is used to cut a piece of cloth (or paper) so that the blades move longer on the cloth (or paper) 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. 3.3. 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 by keeping the effort arm shorter than the load arm so as to give a large movement to the blade of the oar to push water (i.e., load L) back through a longer distance (i.e., gain in speed). The force of reaction exerted by water on the boat moves the boat forward. Its mechanical advantage is less than 1.

(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. 3.1 (b). The load and effort are in opposite directions and the effort arm is always longer than the load arm. Therefore from eqn. (3.10), M.A. is always greater than 1 and since in an ideal machine M.A. is equal to V.R., hence V.R. is also greater than 1. Thus,

The mechanical advantage and velocity ratio of class II levers are always more than 1.

In other words, a class II lever always acts as a force multiplier i.e., a 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.

Examples: 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, raising the weight of the human body on toes, are the examples of Class II levers. Some of these are shown in Fig. 3.4.

Note: If effort arm of a class II lever is 10 cm and load arm is 4 cm, in ideal case its M.A. = 10 / 4 = 2.5, so its V.R. will also be 2.5. In actual lever, if the efficiency is 0.6, i.e., M.A. - V.R., then the V.R. will remain 2.5, but its M.A. will become 2.5 × 0.6 = 1.5 (i.e., it will decrease from 2.5 to 1.5).

(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. 3.1 (c). The effort and load are in opposite directions and the effort arm is always smaller than the load arm. Therefore from eqn. (3.10), M.A. - 1 and since in an ideal machine M.A. is equal to V.R., therefore, V.R. - 1 for these levers. Thus,

The mechanical advantage and velocity ratio of Class III levers are always less than 1.

With levers of class III, we do not get gain in force, but we get gain in speed, i.e., a larger displacement of load is obtained by a smaller displacement of effort. For example, the blade of a knife moves longer by a small displacement of its handle.

Examples: Sugar tongs, the forearm used for lifting a load (or action of the biceps muscle), fire tongs, foot treacle, knife, a spade used to lift coal (or soil), fishing rod, etc. are the examples of Class III levers. Some of these are shown in Fig. 3.5.

The short handle oar used in a boat race by boatmen acts as a lever of class III. In Fig. 3.6, 'o' represents the positions of different boatmen rowing the boat, sitting near the edge of boat on both sides, one after the other, all facing towards the front. Each boatman sitting on the right side holds the top of the oar longitudinally by his left hand. This end acts as the fulcrum F. He applies the effort E on the handle at some distance from the fulcrum, using his right hand. The blade of each oar pushes the water which acts as load L at the other end of the oar. The oar is raised out of water and the process is repeated. Since effort arm is shorter than the load arm, each oar provides gain in speed. All boatmen row the boat in unison so as to obtain more gain in speed and win the boat race.

Note: If a class III lever has the effort arm 2.5 cm and load arm 10.0 cm, in ideal case its M.A. is 2.5 / 10.0 = 0.25, so its V.R. will also be 0.25. In actual lever, if the efficiency is 0.6 (i.e., M.A. - V.R.), its V.R. will remain 0.25, but its M.A. will become 0.6 × 0.25 = 0.15 (i.e., it will reduce from 0.25 to 0.15).

3.7 Examples of Each Class of Levers as Found in the Human Body

In the human body, we can find the examples of all the three classes of levers. The muscles exert force (i.e., effort) by contraction.

(1) Class I Lever in the Action of Nodding of Head

Fig. 3.7 shows the action of nodding of head. In this action, the spine acts as the fulcrum F, load L is at its front part, while effort E is at its rear part. Thus this is an example of Class I lever.

(2) Class II Lever in Raising the Weight of the Body on Toes

Fig. 3.8 shows how the weight of the body is raised on toes. The fulcrum F is at toes at one end, the load L (i.e., weight of the body) is in the middle and effort E by muscles is at the other end. Thus this is an example of Class II lever.

(3) Class III Lever in Raising a Load by Forearm

Fig. 3.9 shows the action of forearm: the biceps. The elbow joint acts as fulcrum F at one end, biceps exerts the effort E in the middle and load L on the palm is at the other end. Thus this is an example of Class III lever.

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