Michael Malika: MACHINES - Simple Machines

PURPOSE OF STUDY

The study outlines the types of levers- that is, first class, second class and third class levers.

EXPECTED LEARNING OUTCOMES

By the end of this study you should be able to:

State the types of levers,
Explain mechanical advantage and velocity ratio,
Calculate mechanical advantage, velocity ratio and the efficiency of a machine,
Measure the mechanical advantage of an inclined plane,
Describe the operation of pulleys,
Account for the simple operation of the wheel and axle and the gears.

CONTENT

All machines, whether simple or complex, allow a force applied at one place to overcome another force at a different place. Overcoming a force involves doing work. Machines in action work by taking in energy at one end and feeding it out at the other end, perhaps in a different form. So, to understand machines, we must look at them one by one – that is levers, pulleys, wheel, and axle and gears.

A machine is a device which is used to overcome a force called the load. This force is applied at one point and the machine works by the application of another force at a different point. For example, a small effort exerted on the rope of a pulley overcomes the weight of the object being raised by the pulley.

Load is the resistance which a machine overcomes and effort is the force used to operate the machine.

MECHANICAL ADVANTAGE (M.A.)

Force Multipliers and distance multipliers

Machines may be designed either to increase the size of a force or to increase the distance or speed with which something moves. In mechanical machine, the energy in put is supplied by a force called the effort and the energy out put is obtained as the machine is used to do work in moving the load.

Force multipliers

Machines that allow a small effort to move a larger load are called force multipliers. Some examples of force multipliers are:

a crowbar
a wheelbarrow
a nutcracker
a bottle opener

The number of times a machine multiplies the effort is called the mechanical advantage.

Mechanical advantage is the ratio of load to effort or load divided by effort.

A mechanical advantage of greater than one means that the load overcome is greater than the effort. The mechanical advantage of a perfect machine is always the same; that of any given real machine increases slightly with load because useless load becomes negligible as load increases.

Mechanical advantage is a ratio of same units therefore has no units.

M.A. = Load

Effort

Velocity ratio is the ratio of distance moved by effort to the corresponding distance moved by the load. It has no units. A velocity ratio greater than one means that the effort moves further than the load.

V.R. = distance moved by effort________

Corresponding distance moved by the load

The mechanical advantage of a machine is the number of times the load moved is greater than the effort used.

Mechanical advantage [M.A.] = load

Effort

Mechanical advantage is a ratio and so has no units. Force multipliers have a mechanical advantage greater than one 1 [M.A. > 1]

M.A. = Load

Effort

= 20N

= _4_

Thus, the mechanical advantage [M.A.] of a machine can only be found by measurement. The example shows a simple lever with unequal arms being used as a force multiplier. So, the lever has a mechanical advantage [M.A.] given by ^Load/_effort, which is 4.

Activity

Water is drawn from a well as shown.

Effort 2N

Mass = 5 kg

Distance multipliers

Machines that are designed as distance or speed multipliers take a small movement of the effort and multiply it to produce a larger movement of the load. Examples include the human forearm and a fishing rod. In these machines, a small movement by the effort produces a much larger movement of the load.

The bicycle is an example of a speed multiplier. The chain and gear wheels of a bicycle are designed so that for a slow movement of the pedals a much faster rotation of the wheels is produced. Distance and speed multipliers have a velocity ratio less than 1 [V.R. < 1].

Velocity ratio (V.R.) (distance ratio)

When a machine is used to multiply the effort, energy conservation is maintained by making the effort move a greater distance than the load. This is necessary because the energy output from the machine (load x distance the load is moved) cannot exceed the energy input (effort x distance effort moves).

The number of times further the effort moves than the load is called the distance ratio.

It is also called the velocity ratio [V.R.] because an effort which moves further than the load in the same time also moves faster.

Distance or velocity ratio = ______distance moved by the effort______

Corresponding distance moved by the load

v The distance or velocity ratio of a machine has no units.

v The distance or velocity ratio of a force multiplier is always greater than 1.

v The distance or velocity ratio of a machine is always greater than its mechanical advantage.

Finding a distance or velocity ratio

The distance or velocity ratio of a machine can usually be calculated exactly from its design or geometry. For example, calculate the distance moved by the effort and load for the lever shown in figure 3.

Figure 3

20 cm

distance moved

by load 10 cm

60 cm distance moved by

effort

40 cm

V.R. = distance moved by the effort

Distance moved by the load

= 40 cm

10 cm

= _4_

The velocity ratio of the lever is 4.

Efficiency of a machine

Energy conservation demands that the total energy output of a machine must equal its energy input. However, when we measure the energy output as work done on the load by a machine, we find it is less than the energy input.

The work done by a machine against its load (moving, lifting, cutting, twisting etc.) is called its useful work or useful energy output. In a simple mechanical machine we can measure this useful energy output as the load multiplied by the distance the load is moved by the machine.

The machine also does work against frictional forces and sometimes does work in moving itself. For example, in a pulley, which we will discuss later, work is done in lifting the moveable pulley block and hook to which the load is attached.

The work done against friction coverts input energy into wasted heat energy and a little noise energy (which eventually becomes heat energy.) the energy equation now looks like this:

Energy input = useful energy output + wasted energy output

As a machine wastes some of its input energy, it is not completely efficient at converting the input energy into the desired output form. Therefore, we measure the efficiency of a machine (usually stated as a percentage) by the following ratio:

Efficiency = useful energy output x 100%

Energy input

The efficiency of a mechanical machine can be calculated from the input and output work done:

Useful work output = load x distance load is moved

The input work = effort x distance effort moves

So that

Efficiency = load x distance load is moved x 100%

Effort x distance effort moves

Looking at this formula, we can see that it contains the relations for M.A. and V.R. It is often useful to be able to calculate the efficiency of a machine from values of its M.A. and V.R. So we have the alternative formula of efficiency of a machine:

Efficiency = M.A. x 100%

V.R.

Efficiency of the machine is the ratio of the work done by the machine (i.e. work output) to the work put into the machine or effort (work input). Efficiency is always expressed in percentage. All real machines have an efficiency of less than 100% due to useless load. Perfect machines are 100% efficient.

Consider the diagram below:

Effort

Load

Efficiency = work output x 100%

Work input

We know that, work output occurs where the load is and work input happens at the effort.

Thus, Efficiency = work done by the load x 100%

Work done by the effort

And we know that work is equal to force x distance moved in the direction of the force

Therefore,

Work = force x distance in the direction of the force

W = f x d

Now the type of a force on work output is the load and the force on work input is the effort.

Considering the formula of work above, it follows that:

Efficiency = load x distance moved by the load x 100%

Effort x distance move by effort

= load x distance moved by the load x 100%

Effort distance move by effort

The formula for Mechanical advantage (M.A.) is = Load

Effort

And that of velocity ratio (V.R.) is = distance moved by effort

Distance moved by the load

Considering the above formula for efficiency which is:

Efficiency = load X distance moved by the load x 100%

Effort distance move by effort

Analysing the formula above, we discover that;

M.A. = load

Effort

Distance moved by the load is the same as the reciprocal of V.R. i.e. ¹/_V.R.

Distance move by effort

Therefore, efficiency = M.A. x ¹/_V.R.= M.A. x 100%

V.R.

Efficiency is expressed as a percentage (%).

Example

A mechanic at the garage uses a pulley machine with a velocity ratio of 6 to raise an engine out of a vehicle with a force of 500 N. The engine, which has a weight of 2,800N, is raised a vertical distance of 1.5m by the machine.

a) The work done by the mechanic

b) The useful work done by the pulley machine

c) The mechanical advantage of the machine

d) The efficiency of the machine

Solution

a) using the formula for V.R. rearranged, we have

V.R. = distance moved by effort

Distance the load is moved

Distance effort moves = V.R. x distance the load is moved

(The V.R. of the pulley machine tells us that the effort will move 6 times further than the load is raised)

\ distance effort moves = 6.0 x 1.5m = 9.0 m

Now the work done by the mechanic, i.e. by the effort, is given by the effort x distance effort moves

\ work done (effort) = 500 N x 9.0m = 4,500 Nm = 4,500 J

b) The useful work done by the pulley machine = force to overcome the load x distance load is raised.

\ Useful work done = 2,800N x 1.5 m = 4,200 Nm = 4,200 J

c) M.A. = load = 2,800 = 5.6

Effort 5,00

d) the efficiency of the machine is given by two methods:

Method 1

Efficiency = useful output work x 100%

Input work

= 4,200J x 100%

4,500J

= 93%

Method 2

Efficiency = MA x 100%

V.R.

= 5.6 x 100%

= 93%

Efficiency and friction

Friction between moving parts in a mechanical machine is the main cause of its wasted work which produces unwanted heat energy. Reducing friction in a machine improves it efficiency and saves energy. Friction between two surfaces may be caused by roughness of the surfaces in which points or projections from one surface bump into similar projections from the other surface. These projections may be so small that they can be seen only with a microscope. Obviously, if the surfaces can be made smoother then they will be able to slide over each other more easily.

However, some surfaces which are very flat and smooth, such as the metal surfaces of a bearing in a car engine, will’ stick’ together even though they are as smooth as a polished mirror. The frictional force between these surfaces can come very close together and the molecules in one surface are strongly attracted to molecules in the other surface. So strong is this mutual attraction between the two surfaces that at the points of closest contact the surfaces are effectively welded together. When they are pulled apart molecules are ‘torn’ from each surface, producing visible damage or scratches. Damage to surfaces in contact as they move against each other is called wear and is said to be caused by friction.

Activity

Explain with examples how friction in a machine can be reduced.

……………………………………………………………………………………………

Reducing friction

Most problems with friction in machines are reduced or overcome in one way or another by keeping the moving surfaces apart. The following list gives most of the methods used to reduce friction in machines:

v Moving surfaces are made as smooth as possible.

v Lubricants such as oil and silicone are used to separate surfaces.

v Machines are moved on rollers and wheels to reduce friction with the ground, rollers and ball bearings are used to separate rotating axles from their mountings.

v Machines are held above the ground by cushions of air (as in the hovercraft) and some bearings use compressed air as an elastic lubricant. Air has the advantage over oil in that, being a gas, it is compressible and therefore has extra useful elastic properties which cushion vibrations in a rotating axle.

v Machines which move through fluids such as water and air are made streamlined in shape to reduce the frictional drag.

LEVERS

A lever is any rigid object which is pivoted about an axis called the fulcrum (F). The load and the effort can be applied on the same side of the fulcrum or on either side.

A lever uses a pivot or fulcrum to transfer the work done by the effort at one point to another point

Activity

In the space below, draw

a) A pair of scissors

b) A human forearm

c) Tongs

In each case, note the following:

a) Where the pivot is

b) Whether the effort or the load is nearer the pivot.

There are three classes of levers, namely; First Class Lever, Second Class Lever and Third Class Lever.

FIRST CLASS LEVER

This type of lever has the pivot between the load and the effort. The scissors and the screwdriver used to remove a tin lid have a velocity ratio greater than 1 and, using the principle of the crowbar, magnify the effort.

Activity

Define the first class levers in your own words.

------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

In first class lever the fulcrum is between the effort and the load. Examples of first class levers include;

- a crowbar

- a craw-hammer

- a pair of scissors

- see-saw

SECOND CLASS LEVER

This type of lever has the load between the effort and the pivot. The wheelbarrow and the bottle opener show how this arrangement also gives a velocity ratio greater than one and magnifies the effort.

Activity

Think of other types of levers that fall under this class. Write them in the space below.

What about the hoe and axe? Do you agree or disagree that they are second class levers? Give your reasons in the space below.

In second class lever, the load is between the fulcrum and the effort. Examples of second class levers include:

- oar in water

- bottle opener

- nut cracker

- wheelbarrow

THIRD CLASS LEVER

These have the effort between the load and the pivot. Fishing rods are designed like this so that a small movement of the effort produces a magnified movement of a load. In the case of the tongs, the force applied to the load is much smaller than the effort and so allows for a fragile object to be held very gently.

Activity

You are provided with food and are to use a knife and fork. Explain why the food on the fork is a type of third class lever.

In third class lever, the effort is between the load and the fulcrum. Examples of third class levers include:

- forceps

- spade

- fishing rod

- biceps muscles and the forearm

- tongs

MECHANICAL ADVANTAGE OF A LEVER

The mechanical advantage of a lever can best be understood if we apply the principle of moments of forces about the fulcrum. Consider the diagram of a crowbar below:

Load arm Effort

Effort arm

load

For the effort to overcome the load, we can apply the principle of moments, which is:

Sum of clockwise moments = sum of anticlockwise moments

For the lever shown in the diagram above:

Effort x effort arm = load x load arm

Thus, effort arm = load

Load arm effort

But we know that M.A. is = load

Effort

Therefore, the M.A. of a lever is = effort arm

Load arm

Note that the moments are taken about the fulcrum or pivot in all the three classes.

Consider the diagrams below:

First class lever (M.A.)

l e

Load fulcrum effort

Second class lever (M.A.)

l e

fulcrum load effort

third class lever (M.A.)

e l

fulcrum effort load

For each class the effort arm is represented by the letter e while the load arm is represented by the letter l.

First class lever

The mechanical advantage of first class lever may take any positive value because ^e/_l can become any fraction. But for a larger M.A., the load arm must be smaller than the effort arm.

Second class lever

The mechanical advantage of a second class lever is always greater than the unity because the effort arm e is always greater than the load arm l.

Third class lever

The mechanical advantage of a third class lever is always less than the unity because the value of the effort arm is always less than the load arm.

PULLEYS

A pulley is a grooved wheel which is able to turn about an axis fixed in a frame. In a pulley a rope, belt or chain passes over the pulley in a groove. The pulley is able to transmit motion.

Pulleys are used to lift weights or pull objects. They can also be used to change the direction of the applied effort.

Pulleys are used to change the direction of a force and to gain a mechanical advantage greater than 1.

SINGLE FIXED PULLEY

Below is a diagram showing a single fixed pulley. Such a pulley is used to raise small objects such as a goal post, flag pole etc.

A fixed pulley is one with a fixed support which does not move with either the effort or the load. The pulley itself should turn on its axle as freely as possible for maximum efficiency. Figure 4 shows a single fixed pulley.

This pulley is used to change the direction of the effort force E from a downward pull to an upwards lift. The tension T in the string or rope applies the upwards force to the load L. it is often easier to pull a rope downwards than to lift a load upwards.

The V.R. of a single fixed pulley must be exactly 1 as the load will rise by the same distance as the effort moves.

The M.A. will be almost 1, there being only a small amount of work wasted against friction on the pulley bearing and in lifting the weight of the rope.

An effort is applied to one end of the rope and a load to the other end. On application of the effort, a tension is created in the rope so that the load can be raised. If we neglect frictional force, the tension in the string is equal to the applied effort. Since it is a single fixed pulley, the effort is also equal to the load being raised.

Consider the diagram.

Effort

Load

Thus, effort (E) = tension (T)

And, effort (E) = load (L)

But we know that M.A. = load

Effort

Now since effort = load

Therefore, M.A. = 1.

It follows that the M.A. of a single fixed pulley is 1.

The velocity ratio of a single fixed pulley is unity because the distance moved by the effort is equal to the distance moved by the load.

SINGLE MOVABLE PULLEY

A single moving pulley gives a V.R. of 2. This can be seen in the figure.

For any distance the load is raised, there are two lengths of rope equal to that distance to be pulled upwards by the effort. So, both ropes supporting the load must be shortened by the distance the load is raised.

The mechanical advantage is found by measurement, but we can see that the upwards lifting force is shared equally between two upwards forces, the effort (E) and the tension (T) in the rope at the other side of the pulley.

Consider the diagrams of a single movable pulley below:

T T

Load

(a)

T T

Load

(b)

Total upward force needed = load + the weight of the moving pulley etc.

So, the effort needed = ½ (load + weight of pulley)

If the load is heavy compared with the pulley and frictional force, then the effort needed will be roughly half the load and the M.A. nearly 2. In other words, a single moving pulley can be used to magnify an effort force by almost 2.

A single moving pulley is often combined with a single fixed pulley to provide a simple machine with a downwards effort and a V.R. of 2.

Mechanical advantage of a single movable pulley

In diagram (a), as the effort is applied at the free end E, a tension is created in the rope which is the same throughout and the tension is equal to the effort (E).

Effort = T

Since the load is supported by two sections of the rope,

Therefore L = 2T

Therefore, mechanical advantage, which we know is load, is

Effort

M.A. in terms of tension T = ^2T/_T= 2

Velocity ratio of a single movable pulley

Diagram (b) helps us to understand velocity ratio. The fixed will is there in diagram (b) just to change the direction of the effort.

If the effort moves a distance of x, then the supporting ropes will each move a distance which is half of x since they are two. i.e. ¹/₂x.

The load will move a distance which is equal to ¹/₂x since it is supported by two ropes

Therefore if the effort move through a distance x, the load moves ¹/₂x

Velocity ratio = distance move by effort

Distance moved by load

= __x__

¹/₂x

= x ¸ ¹/₂x.

= 2

A BLOCK AND TACKLE SYSTEM OF PULLEYS

Pulleys are usually used in sets of two or more in order to gain a higher V.R. and M.A. Two sets of pulleys are used, one fixed and one moving. The pulleys are mounted side-by-side in a block or frame, and the apparatus of pulleys and ropes is generally called the tackle. The whole system or machine is known as a block and tackle. Many large machines use the block and tackle pulley systems, for example cranes and lift mechanisms.

So that we can see more clearly where the ropes go, we usually draw the pulleys below each other in each block. The lower, moving pulley block is supported by five ropes. To raise the load by 1 metre will require 5 metres of rope to be pulled out of the machine by the effort and so the V.R. is exactly 5.

Consider the diagram below:

Effort

Load

Experiment

To measure the M.A. and efficiency of a block and tackle pulley system

In a perfect machine, the four tension forces T which support the load would take exactly a quarter of the load each. The effort force providing the tension would also be ¹/₅ of the load, and so the mechanical advantage would be exactly 5. But we can only find the M.A. of a particular machine by measuring the effort needed to overcome or lift a particular load.

Using a block and tackle with a V.R. of 5 and slotted masses from 1 to 10 kg, a suitable spring balance would measure over the range 0 to 30N.

v Record readings of the spring balance for the effort needed to just raise the load for loads from ½ kg to 10 kg.

v Make a table of results as in the following table:

Mass of load/kg	Load /N	Effort /N	M.A. =^L/_E	E = ^M.A/_V.R. x 100%
0.5	5	2.5	2.0	50%
1	10	3.5	2.9	72%

v Plot a graph of M.A. against load and of efficiency against load (see figure 7).

Mechanical advantage against load

Figure 7

M.A.

⁶

₅

₄

0 load /N 100

Efficiency against time

Efficiency (%)

¹²⁰

₁₀₀

₈₀

⁶⁰

₄₀

₂₀

0 load /N 100

Some typical results, shown in the graphs, indicate an increase in M.A. and efficiency as the load increases. This is to be expected because at high loads, the weight of the moving pulley block and frictional forces are very small compared with the load. Note that the V.R. which is constant at 4 in this example, limits the M.A. to a maximum which is always less than the V.R.

As the effort is applied at the free end of the rope, a tension created in all the four ropes supporting the movable block. The tension so created is equal to the effort applied at the free end. Hence the total tension in the ropes supporting the movable block is equal to the load.

Since the load is equal to the total tension in the ropes supporting the movable block and also the tension in each section of the rope is equal to the effort: i.e.

Effort = Tension

Load = total tension in the four sections of the ropes

It follows that:

Load = effort x number of rope sections supporting the movable block

Because the number of rope sections supporting the movable block is equal to the number of pulleys in the system, it follows that:

Load = effort x number of pulleys in the system

Mechanical advantage of the block and tackle system of pulleys

We know that M.A. = load

Effort

In the block and tackle system of pulleys load is given by the formula:

Load = effort x number of pulleys in the system

It follows that M.A. = effort x number of pulleys in the system

Effort

M.A. = number of pulleys in the system

Velocity ratio of the block and tackle system of pulleys

We know that:

V.R. = Distance moved by effort

Distance moved by load

Now in a block and tackle system of pulleys, if the effort moves through a distance of x, ropes are shortened depending on the number of rope sections supporting the movable block. Now in the system of pulleys above, the number of rope sections are 5, hence each section of the rope is shortened by a distance of ^x/₅ and the load moves a distance of ^x/₅upwards.

V.R. = ____x____ = 5

^x/₅

therefore = number of pulleys in the system

Hence in a perfect block and tackle system of pulleys M.A. = V.R.

Examples

A block and tackle system of pulleys has a V.R. of 4. If a load of 200N is raised by using a force of 75N, calculate the mechanical advantage and efficiency of the system.

M.A. = load

Effort

= 200

= 2.67

Efficiency = M.A.

V.R.

= 2.67

= 67 %

A block and tackle pulley system consists of 5 pulleys. If the efficiency of the system is 80%, what load will be raised by an effort of 100N?

Since the system has 5 pulleys, the V.R. is 5

We know that:

Efficiency = M.A. X 100

V.R.

80% = M.A. x 100

5 x 80 = 100 M.A.

M.A. = 5 x 80

100

= __4__

We know that: M.A. = load

Effort

4 = load

100

Load = 400N

INCLINED PLANE

Inclined plane is a plane surface at an angle to the horizontal. It is easier to move an object up an inclined plane than to move it vertically upwards.

You have already seen the work done in lifting a load depends on the vertical distance moved. An inclined plane is a slope or ramp which allows a load to be raised more gradually and by using a smaller effort then if it were lifted vertically upwards. Bolts and screws are based on the principle of an inclined plane. Inclined planes, bolts and screws are machines which magnify the effort, i.e. they have a M.A. greater than 1.

The velocity ratio of an incline plane

Consider the diagram:

Effort

Height

Length of plane

B O

In the figure, the effort (E), pushing up the inclined plane or slope, moves a distance (d) along the slope. The load (L) is raised a vertical height (h) against gravity.

V.R. = distance effort moves along the slope = d

Vertical height load is raised h

Measuring the M.A. of an inclined plane

v Find the weight of a small trolley by hanging it on a 0 to 10N spring balance.

v Now pull the same trolley up a sloping ramp or runway using the spring balance. Read on the spring balance the effort force which is just large enough to keep the trolley moving up the slope at a constant speed.

v Change the gradient of the slope and investigate how the effort, MA and efficiency are affected.

v Do the calculations in the same way as for the block and tackle pulley system.

We know that:

V.R. = Distance moved by effort

Distance moved by load

\ V.R. = length of the plane

Height of the plane

= OA

In an inclined plane, the work done by effort is equal to the work done by the load

Therefore, effort x length of the plane = load x height of the plane

But M.A. = load

Effort

M.A. = Length of plane

Height of plane

Example

A loaded wheelbarrow of weight 800N is pulled up an inclined plane by a force of 150N parallel to the plane. If the plane rises by 50 cm for every 400 cm distance measured along the plane, find the velocity ratio, mechanical advantage and efficiency of the plane.

V.R. = length of plane

Height of plane

= 400

= __8__

M.A. = load

Effort

= 800

150

= 5.3

Efficiency = M.A. x 100

V.R.

= ^5.3/₈ x 100

= 66.25%

THE SCREW JACK

This is a system in which a screw thread is turned to raise a load (e.g. a car jack). The pitch is the distance between each thread on the screw. In this system, one revolution of the handle (effort moves in circle radius) raises the load by the pitch.

The distance between adjacent threads (ones next to each other) is called the pitch of the thread. When the bolt is turned through one full turn it moves up or down a distance equal to the pitch of its thread. The threaded bolt can be pulling or lifting a load as it turns. The distance the load is moved for each turn of the bolt equals the pitch of the thread. The distance round one turn of the thread, along the ramp, is the corresponding distance the effort moves.

Velocity Ratio of the Screw Jack

Consider the diagram:

We know that:

V.R. = Distance moved by effort

Distance moved by load

\ V.R. = circumference of the circle

Pitch of the screw

= 2 Π r

Example

A screw jack with a pitch of 0.2 cm and a handle of the length 50 cm is used to lift a car of weight 1.2 x 10⁴ N. if the efficiency of the screw is 30% find:-

The velocity ratio and mechanical advantage of the machine
The effort required to raise the car

We know that:

V.R. = Distance moved by effort

Distance moved by load

\ V.R. = circumference of the circle

Pitch of the screw

= 2 Π r

= 2 x ²²/₇ x ⁵⁰/_0.2

V.R. = 1517.4

We know that:

Efficiency = M.A. x 100%

V.R.

30 = M.A. x 100

1517.4

M.A. = 30 x 1517.4

100

= 471.4

Also we know that M.A. = load

Effort

471.4 = 1.2 x 10⁴

Effort

Effort = 120000

4714

= 25.5N

WHEEL AND AXLE

A car steering wheel provides a good example of the wheel and axle principle.

radius of axle Radius of wheel

Effort

Load

R = radius of wheel; r = radius of axle

If you had no wheel on the axle and you tried to turn the axle by hand you would fail because you could not apply a strong enough effort to it. The steering wheel allows you to see a small effort to overcome a large load. A steering wheel has a M.A. greater than 1 and so magnifies your effort.

The V.R. of the wheel and axle can be found from the radii R and r as shown in the figure above. The distance moved by the effort in turning the wheel round once is the circumference of the wheel, 2 Π R. The distance moved by the load acting around the circumference of the axle is 2 Π r

Therefore, V.R. = distance moved by the effort

Distance moved by the load

= 2 Π R.

2 Π r

= ^R/_r

Activity

1. You are given the radius of a wheel as 1.5m and the axle being 0.5m. Determine its velocity ratio.

2. in your area, measure the wheel and axle on the parts of

a. the well for your drinking water

b. The hammer mill where you take your maize for grinding.

Example

A wheel and axle of efficiency 80 % is used to raise a load of 2000N. If the radius of the wheel is 50 cm, and that of the axle is 2 cm, calculate:

The velocity ratio and mechanical advantage of the machine
The effort required to overcome the load

We know that:

V.R. = Distance moved by effort

Distance moved by load

V.R. = ^R/_r

= ⁵⁰/₂

= 25

We know that:

Efficiency = M.A. x 100%

V.R.

80 = M.A. x 100

M.A. = 80 x 25

100

= 20

Also we know that M.A. = load

Effort

20 = 2000

Effort

Effort = 2000

= 100N

The Mechanical Advantage of, velocity Ratio and Efficiency of a Hydraulic Press

Consider the diagram:

r y

On the application of a small downward effort E on the piston of radius r, the load piston of radius R lifts the load L

Applying the principle of transmission of pressure in liquids, the pressure on the effort piston equals the pressure on the load piston.

Therefore, ________effort ________ = _________load _______

Area of the effort piston area of the load piston

or load = area of the load piston

Effort Area of the effort piston

But we know that M.A. = load

Effort

Thus, M.A. = area of the load piston

Area of the effort piston

M.A. = Π R²

Π r²

= (^R/_r)²

Neglecting friction, the work done by the effort E is equal to the work done on the load L. So if the effort piston moves a distance of x downwards, and the load piston raised up by a corresponding distance y, it follows that:

Effort x distance x = load x distance y

So distance x = load

Distance y = effort

x = L

y E

But ^x/_y velocity ratio and ^L/_E is mechanical advantage which is equal to (^R/_r)²

Therefore, V.R. = (^R/_r)²

SUMMARY

· A lever is a simple machine which uses a pivot to transfer work done by effort.

· There are three classes of lever- first class, second class and third class.

· First class levers have the pivot between the load and the effort- e.g. screwdriver, scissors.

· Second class levers have the load between the effort and the pivot- e.g. wheelbarrow.

· Third class levers have the effort between the load and the pivot – e.g. fishing rods.

· Mechanical advantage (M.A.) is the number of times a machine multiplies the effort.

· Mechanical advantage = load

Effort

· Mechanical advantage is a combination of force multipliers and distance multipliers.

· Force multipliers allow a small effort to move a larger load.

· Distance multipliers take a small movement of the effort and multiply it to produce a larger movement of the load.

· Velocity ratio (V.R.) is the number of times the effort moves than the load.

· Velocity ratio = distance moved by effort

Distance load moves

· Efficiency of a machine shows if a particular machine is useful or not by comparing the energy output to energy input.

· Efficiency = energy output x 100%

Energy input

· Effective machines have a low friction level.

· There are a number of ways of reducing friction, such as lubricating or polishing surfaces, use of rollers etc.

· Pulleys are used to change the direction of a force and to gain a mechanical advantage greater than 1.

· A single fixed pulley is one which ahs a fixed support which does not move with either the load or the effort.

· A single moving pulley gives a velocity ration of 2.

· A block and tackle is a combination of a single fixed pulley and a single moving pulley.

· Inclined plane/ slope/ ramp allow a load to be raised up more gradually using a smaller effort than if it were lifted vertically upwards.

· In bolts and screws, the distance between adjacent threads is called pitch.

· A bolt turned one full turn moves up or down a distance equal to the pitch and can be used in pulling or lifting a load.

· In the wheel and axle, it becomes easier to turn the axle using the wheel with very little effort.

· Gears are rigidly fixed wheels to axles, designed to slow down the speed of rotation and to magnify the effort force.

SELF-CHECK EXERCISE

Show by drawing the position of the fulcrum, effort and load in each of the three classes of levers.

It may be said that machines make work easier, but do not make it any less. If you have, for example, a lifting job to do, you may use a pulley system to do it

Effort

Load

Calculate the work done in raising the load directly if it weights 300N and has to be raised through a height of 20m.
Calculate the work you would do in raising yourself through the same height if you weighed 600N.
Calculate the work done by the effort of 450N if moved 20m.
What is the efficiency of the machine?

A boy of mass 50 kg runs up a hill of vertical height 300m in 20 minutes. Using g = 10N/ Kg, calculate:

The average vertical force he uses to lift himself up the hill,
The work he does climbing the hill,
His average power.

The electric motor in the figure below completely lifts the box in 5s at a steady speed. If the box weighs 100N, calculate:

a. the work done by the motor,

b. The power of the motor.

2 m

The figure shows a wire of Tanzanian Railway’s Electrification system, being held taut by a load L and a pulley system P.

Load

a. By what factor is the force multiplied?

b. What is the purpose of pulley 1?

c. Why are pulleys used at all?

d. If the load L is 2,000N, what is the tension in the wire W?

A block and tackle pulley system with a velocity ratio of 5 and 60% efficiency is used to lift a load of mass 60 kg through a vertical height of 2 metres.

a) What effort must be exerted?

b) How much work is done in lifting the load?

c) How much work is done by the effort?

(Assume that the acceleration of freefall is 10m/s².)

In reality, a block and tackle or compound pulley system would be used. The system shown in the diagram below has five ‘lifting’ strings.

Effort = 150N

Load = 600N

a. Find the velocity ratio.

b. Calculate the mechanical advantage and

c. Efficiency of the system.

With the sling round the crate 60 kg, the pulley system shown below is used to lift the crate through a height 2 metres

a. What is the velocity ratio of the pulley system

b. What effort would have to be applied to the pulley system, if it were 100% efficient, in order to raise the crate?

c. If, in fact, the pulley system is only 80% efficient, how much work is needed to raise the crate through the 2 m, using the pulley system

d. Give one reason why the pulley system is not 100% efficient.

(take g = 10 N/kg)

Effort

Crate

Labels: MACHINES

Michael Malika

Friday, July 22, 2011

MACHINES - Simple Machines

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