Wheel And Axle

Wheel and axle it consist of a wheel and an axle mounted and joined together so they

have the same axis of rotation

Whereby:

Circumference of axle = Load distance (Ld)

Circumference of wheel = Effort distance (Ed)

Radius of wheel = R

Radius of axle = r

For wheel and axle

❖

푳

푴. 푨 =

푬

푬풅

푳풅

❖

ퟐ흅푹

ퟐ흅풓

푹

풓

푽. 푹 =

=

푴.푨

풆풇풇풊풄풊풆풏풄풚 =

풙 ퟏퟎퟎ%

푽.푹

Example

A wheel and axle has a velocity ratio of 6. Determine the radius of the wheel, if the

radius of the axle is (a) 5 cm (b) 8 cm (c) 12 cm

Data given

Velocity ratio, VR = 6

Radius of the wheel, R = ?

Solution:

(a) If radius of axle, r = 5 cm

But: 푽푹 = ^푹

→ 푹 = 푽푹 풙 풓 = ퟔ풙ퟓ = ퟑퟎ 풄풎

풓

(b) If radius of axle, r = 8 cm

But: 푽푹 =^푹

→ 푹 = 푽푹 풙 풓 = ퟔ풙ퟖ = ퟒퟖ 풄풎

풓

(c) If radius of axle, r = 12 cm

But: 푽푹 = ^푹

→ 푹 = 푽푹 풙 풓 = ퟔ풙ퟏퟐ = ퟕퟐ풄풎

풓

Individual Task – 6

A wheel and axle with an efficiency of 90% is used to raise a load of 10000 N. the

radius of the wheel is 40 cm while radius of the axle is 5 cm. find Velocity ratio,

Mechanical advantage and Effort (ANS: V.R = 8, M.A = 7.2, E = 1388.9 N

)

Uses of wheel and axle in daily Life

•

Applied in riding a bicycle

•

Used to fetch water in a well

Hydraulic Press

Hydraulic press multiplies an applied effort using the pressure of a liquid or gas. This

allows the lifting of a heavy load by applying little effort

Assume there is no friction force (By principle of transmission of pressure in

liquids, the pressure on effort piston equal to that on the load piston.)

That is

P₁= P₂

푭

But: P =

푨

Whereby: F₁= E = P₁A_1,F₂= L = P₂A₂

푷 푨

ퟐ

푳

Since M.A =

=

, 풃풖풕 푷_ퟏ= 푷_ퟐ

푬

푷 푨

ퟏ ퟏ

푨

ퟐ

풕풉풖풔 푴. 푨 =

푨

ퟏ

Whereby: A₂= πR²and A₁= πr²

푹^ퟐ

∴ 푀. 퐴 =

풓^ퟐ

•

ퟐ

푹

Thus in practical cases, the mechanical advantage is less than

due to

ퟐ

풓

frictional forces in the hydraulic press .Therefore the efficiency of a

hydraulic press is less than 100%

Velocity ratio of a hydraulic press

•

Assuming that, the hydraulic press is frictionless then

work output = work input

•

That is

L x Ld = E x Ed----------------------- divide by E and Ed both sides

푬풅

푳풅

푳

=

푬

•

But 푀. 퐴 = ^퐿

ퟐ

푹

=

ퟐ

풓

퐸

ퟐ

푹

풓

∴ 푽. 푹 = 푴. 푨 =

(if the hydraulic press is 100% efficient)

ퟐ

❖

푳

푴푨 =

푬

ퟐ

❖

ퟐ

풓

푴푨

❖푬풇풇풊풄풊풆풏풄풚 =

풙 ퟏퟎퟎ%

푽푹

Example

The diagram below shows a hydraulic press being used to lift a container weighting

100000 N

Radii of the effort and load piston are 20 cm and 50 m respectively, if the efficiency of

the hydraulic press is 90%. Determine

a) Velocity ratio

b) Mechanical advantage

c) Minimum Effort

d) The distance the container raised through if the effort piston pushed through 1 m

Data given:

Load, L = 100000 N

Radius of Load piston, R = 5 m

Radius of effort piston, r = 20 cm = 0.2 m

Efficiency, Eff = 90%

Velocity ratio, VR = ?

Mechanical advantage, MA =?

Minimum effort, Emin =?

Distance moved by effort, Ed = 1m

Distance moved by load, Ld =?

Solution

ퟐ

a)

ퟓ

ퟐퟓ

푹

=

V.R =

= ퟔퟐퟓ

=

ퟐ

풓

ퟎ.ퟐ

ퟎ.ퟎퟒ

b) From Efficiency = ^푴.푨× ퟏퟎퟎ% ..............................make MA the subject

푽.푹

)

∴MA = ⁽

=

^{ퟗퟎ 풙 ퟔퟐퟓ}= ퟓퟔퟐ. ퟓ

풆풇풇풙푽.푹

ퟏퟎퟎ

퐿

c) From: MA =

-

-------------- make E the subject

퐸

ퟏퟎퟎퟎퟎퟎ

푳

=

= ퟏퟕퟕ. ퟕퟖ 푵

∴ 푬 =

푴푨

ퟓퟔퟐ.ퟓ

d)

From: 푉푅 = ^퐸푑

-

------------ make Ld the subject

퐿푑

퐸푑

1

∴ 퐿표푎푑 푑푖푠푡푎푛푐푒 =

=

= 0.0016 푚

푉푅

625

Uses of Hydraulic Press in Daily Life

o

It lifts heavy loads

In ginneries to compress a lump of cotton into small bales

In industries to form car bodies into the required shapes

Class Assignment

1. Determine the effort required to lift a load of 100 N using:

(a) Using a single fixed pulley

(b) A single movable pulley

(c) Combination pulley system made up of 5 pulleys

(d) Compare your results, which is the best pulley to use in this case? Why is it

the best?

2. A machine has a velocity ratio of 5 and is 80% efficient .What effort would be

needed to lift a load of 200 N with the aid of this machine?

3. While lifting a load of 200 N using a lever, an effort of 80 N moved through a

distance of 20 cm to lift the load through a distance of 4 cm

Calculate:

(a) The mechanical advantage

(b) The velocity ratio

(c) The efficiency of the machine

4. A block and tackle system consisting of a 5 pulleys is used to raise a load of

400 N through a height of 10 cm .If the work done against friction is 100 J ,

Calculate:

(a) The work done by the effort

(b) The efficiency of the machine

(c) The effort applied

5. If an effort of 15 N is needed to lift a load of 150 N with a three pulley, what is

the efficiency of this machine?

6. (a) In a pulley system , a load of 700 N requires an effort of 100 N to raise it

.What is the mechanical advantage

(b) If this effort moves through a distance of 10 m and the load is then moved

up a distance of 2 m, calculate the velocity ratio and the efficiency of the

machine

7. A machine requires 6000 J of energy to lift a mass of 55 kg through a vertical

distance of 8 m. Calculate its efficiency

8. A pulley system has a velocity ratio of 3. Calculate the effort required to lift a

load of 600 N, if the system is 75% efficient

9. A machine of efficiency 75% lift a mass of 90 kg through a vertical distance of

3 m. Find the work required to operate the machine.

10. A hydraulic machine has a piston P of cross – sectional area 5 cm²and Q of

cross – sectional area of 50 cm².Find the velocity ratio of the system

11. If the VR of a hydraulic machine is 441 and the effort distance moved by the

effort piston is 7 m, Calculate the load distance

12. A load of 30 N is raised 2m when an effort of 10 N moves through 8 m.

Calculate the mechanical advantage and velocity ratio of the machine

13. A 120 N load is pushed up a straight smooth inclined plane. The angle

between the inclined plane and the horizontal is 30⁰.The load is moved a

distance of 5 m along the inclined plane. Calculate:-

a) The vertical height the load is moved

b) The velocity ratio of the inclined plane

c) If the machine has an efficiency of 75% ,calculate the effort force

required

14. The frictional force in a car jack is great. Explain why this must be so?

15. What do you understand by the efficiency of a machine? A system of levers

with a velocity ratio of 25 overcomes a resistance of 3300 N when an effort of

165 N is applied to it ,calculate:

i.

Mechanical advantage of the system (ANS: 20 )

ii.

Its efficiency

(ANS: 80% )

16. Define work and power .Name and define the SI unit of power .By using a

block and tackle a man can raise a load of 720 N by an effort of 200N .Find:

a) The mechanical advantage of the method (ANS: 3.6)

b) The man’s useful power output if he raises the load through 10 m in 90 s

( ANS: 80 w)

17. Draw a diagram of a single – string pulley system with a velocity ratio of 4

18.

19.

20.

21.

A

father exerting a force of 300 N was able to push a wheelbarrow with a load

of mass 120 kg. Calculate the number of times the wheel barrow multiplied the

father’s force ( g = 9.8 N/kg)

A

is used to raise a load of 60 N, what is the efficiency of the machine?

(ANS: Efficiency 75%)

A

lower movable block. What load can be lifted by an effort of 200 N if the

efficiency of the machine is 60 %. (ANS: L = 600 N)

A

(

ANS: M.A = 3.92)

machine, the load moves 2 m when the effort moves 8 m. If an effort of 20 N

block and tackle system has 3 pulleys in the upper fixed block and two in the

wheel and axle is used to raise a load of 280 N by a force of 40 N applied to

the rim of the wheel. If the radii of the wheel and axle are 70 cm and 5 cm

respectively. Calculate the M.A, V.R and efficiency

.

(ANS: M.A = 7, V.R = 14, Efficiency = 50 % ).

22.

A

man uses an inclined plane to lift a 50 kg load through a vertical height of 4.0

m. The inclined plane makes an angle of 30⁰with the horizontal. If the

efficiency of the inclined plane is 72 % ,calculate:-

(a) The effort needed to move the load up the inclined plane at a constant

velocity.

(ANS: Effort = 347.2 N)

(b) The work done against friction in raising the load through the height of

4.0 m.(take g = 10 N/kg)

(ANS: Wd = 777.6 J)

23. A car weighing 1 600 kg is lifted with a jack – screw of 11 mm pitch .If the

handle is 28 cm from the screw , find the force applied. (ANS: E = 10 N)

24. In the figure below, a crowbar was used to lift a 600 N Load placed 1.2 m away

from the turning point. An effort was exerted at a distance of 3m from the

turning point

(a) Calculate the value of the velocity ratio ( ANS: V.R = 0.4)

(b) Assume the machine was frictionless and weightless ,find the value of

the load

(ANS: L= 240 N)

25. A machine used to lift a load to the top of a building under construction has a

velocity ratio of 6. Calculate its efficiency if an effort of 1 200 N is required to

raise a load of 6000 N .Find the energy wasted when a load of 700 N is lifted

through a distance of 3 m.

26. In order to lift a car of weight 5000 N through a vertical height of 200 m, a 700

m long inclined plane is used. Calculate:

i. Velocity ratio of the simple machine

(

ANS: 3.5)

ii. The effort applied

(

ANS: 1428.57 N)

iii. The mechanical advantage of the simple machine (ANS: 3.5)

27.

A

Crate of soda of mass15 kg was raised through a certain vertical height of a

plane inclined at 30⁰with the horizontal .Assume that the plane is frictionless

and g = 10 N/kg. Determine:

a)

Velocity ratio of the simple machine

(ANS: V.R = 2)

b) The effort applied

(ANS:E = 75 N)

c) The mechanical advantage of the simple machine (ANS: M.A = 2)

28. In a hydraulic press ,the small piston has an area of 0.2 m²while the large

piston has an area of 1.5 m².The effort applied on the small piston is 250 N.

Calculate:

(a) The mechanical advantage of the machine

(ANS: 7.5)

(b) The maximum load that the machine can lift

(ANS:1875 N)

(c) If the small piston (effort piston) moves inwards by 3 m, calculate the

distance by which the load is lifted

(d) What is the velocity ratio of the machine

(e) Calculate the efficiency of the hydraulic press

(

ANS: 0.4 m)

(ANS: 7.5)

(ANS: 100%)

29. An experiment was carried out to investigate the performance of a single –

string pulley system with a velocity ratio of five. The following results were

obtained:

Load(N) 50 100 200 300 400 500 600

Effort (N) 30 45 65 85 105 125 145

Plot a graph of effort against Load, Hence find:

a) The effort

(ANS: 115 N)

b) The mechanical advantage and

( ANS: 3.9)

c) The efficiency corresponding to a load of 450 N ( ANS: 78%

)

30.

Draw a diagram of a single – string pulley system with a velocity ratio of 5

.Draw also two other mechanical devices having the same velocity ratio

.Explain in each of the examples ,why you assert that the velocity ratio is

indeed 5.Give one reason common to them all, why the mechanical efficiency

is less than 100% ,and one way of increasing the mechanical efficiency of any

of the machines you have described

31. Describe how a hydraulic press works

32. An inclined plane is 7 m long and its height is 1 m. If the efficiency of the plane

is 70%, find the load which can be moved up the plane by an effort of 150 N

parallel to the plane. (ANS: L = 73500 N)

33. Define the following terms

(a)Pitch of a screw

(b) Fulcrum

(c) Axle

(d) Efficiency of a machine

34. Suppose a screw of radius 0.7 cm has 12 threads per inch .Calculate the

velocity ratio of the screw

(

ANS: V.R = 20.95)

35. Draw a diagram of a single – stringed block and tackle pulley with a velocity

ratio of:

(a) 4

(b) 5

(c) 6

(d) 2

36.

A

box of mass 12 kg is pulled up a straight smooth incline at 30 to the

horizontal , for a distance of 5 m .Calculate the work done (ANS: 300 J)

37. A bicycle has wheels 66 cm in diameter .Its crank wheel has 44 teeth and the

rear sprocket 16 teeth .If the crank radius is 16.5 cm, Calculate the velocity

ratio .Find the efficiency of the bicycle if its mechanical advantage is 0.14

(

ANS: V.R = 0.18, Efficiency = 77% )

38. A screw jack with a pitch of 0.2 cm and a handle of length 50 cm is used to lift

a car of weight 1200 N. If the efficiency of the screw is 30% calculate.

(a) The velocity ratio (ANS: p = 1570)

(b) The mechanical advantage (ANS: M.A = 471)

(c) The effort used to raise the car(ANS: E = 2.55)

39. A driving gear wheel having 25 teeth engages with a second wheel with 100

teeth. A third wheel with 30 teeth on the same shaft as the second ,engages

with a fourth having 60 teeth .Find:

(a) The total velocity ratio

(ANS:8

)

(b) The mechanical advantage of the gear system if its efficiency is 85%

(ANS: 6.8 )

40. Draw a diagram of a single – string pulley system with a velocity ratio of

6.Calculate the efficiency of it if an effort of 1000 N is required to raise a load of

4500 N .Find the energy wasted when a mass of 500 kg is lifted through 2 m

(ANS: Eff = 75%, E = 3333 J)

41. A man uses a rope to haul a packing case of weight 750 N up an inclined

wooden plank of effective length 4.50 m and on to platform 1.50 m high .The

frictional force between case and plank is 200 N. Find:

(a) The effort he must exert on the rope

(ANS: 450 N)

(b) The velocity ratio

(ANS:

3)

(c) The mechanical advantage

(ANS: 1.67)

(d) The useful work done on the packing case in joules (ANS: 1125 J)

42. The machine is used to lift a container weighing 80,000 N. The radius of effort

piston is 50 cm and the radius of a load piston is 5 m. If the efficiency of the

machine is 90%, Calculate the velocity ratio and its mechanical advantage

(M.A) (ANS: V.R = 100, M.A = 90)

43. Define velocity ratio and mechanical advantage .A common windlass is used to

raise a 480 N load of earth from an excavation by the application of an effort of

200 N at right angles to the handle and crank .If the handle is 33 cm from the

axis and the radius of the axle is 11 cm , find:

(a) The velocity ratio

(ANS: 3 )

(b) The mechanical advantage ( ANS: 2.4)

44. A lever 2.4 m long is to be used to raise an object weighing 500 N by applying

an effort of 100 N as shown below. How far from the object should the fulcrum

be placed?

45. Identify the class to which each of the levers given below belongs

A hoe, pair scissors, wire cuter, bottle opener, wheelbarrows

46. The efficiency of a press is given as 75%.If the radius of the load piston is

given as 3 m while the effort piston is 1.5 m, calculate:

(a) the VR

(b) the MA of the press

47. Discuss the mode of action of a hydraulic press

48. Efficiency of a simple machine is never 100%. Why?

49. The pulley system shown below is to be used to raise a load of 150 N

a) Ignoring friction and the weight of the pulleys, what force is required to

raise the load at a constant speed?

b) If the cable (where the effort is applied) is pulled down 60 cm, how far

will the object rise?

50. The effort distance moved by an effort force of 20 N to lift a bag is 10 cm.

I f the M.A of the inclined system is 10, calculate the load distance

51. The figure below shows a pulley system. Find

(a) The velocity ratio of the pulley system

(b) The mechanical advantage, if the system is 80% efficient.

(c) The effort

(d) The work done by the effort in lifting the load through a distance of 0.7 m

(e) How much energy is wasted

52. A box weighing 560 N is pulled along an inclined plane of length 5 m onto a

platform 2 m high with a force of 70 N. Calculate:

a) The velocity ratio of the plane

b) The mechanical advantage of the plane

c) The efficiency of the plane

53. The diagram below shows a box of weight 150 N being pulled at a steady

speed up a ramp by a force of 100 N

If the height of the plane is 2 m above the ground and the length of the

plane is 4 m, Calculate:

(a) The mechanical advantage

(b) The velocity ratio

(c) The work done by the effort

(d) The work done on the load

54.

A

200 kg crate is to be loaded onto the bed of a truck that is 1.4 m above the

ground. A metal ramp 5 m long is leaned against the truck bed and the crate

pushed along it. Neglecting the frictional force:

(a) calculate the force required to push the crate up the incline at a constant velocity

(b) calculate the ma of the incline

(c) determine the efficiency of the machine

55. A trolley is pulled up an inclined plane 2 m high using a force of 4 N. If the mass of

the trolley is 1 kg

(a) What is MA of the plane

(c) Find its efficiency

(b) Find the velocity ratio

56. The figure below shows a box whose weight along the inclined plane is 300 N

Calculate:

(i) The work done in sliding the box up the inclined from P to Q

(ii) The power used if it took 5 minutes to move the box from P to Q