Chapter 4 — The Geometry of Power: Advanced Simple Machines

1. Mechanical Advantage

Machines allow us to use a relatively small effort to produce a larger useful effect.

This is expressed through Mechanical Advantage (MA).

The chapter illustrates this idea using steering systems, cranes, bicycles and gears.

2. Wheel and axle

A wheel and axle consist of:

• a larger wheel;
• a smaller axle attached to its centre.

Both rotate together.

For the ideal wheel-and-axle arrangement:$$MA=\frac{\text{radius of wheel}} {\text{radius of axle}}$$

Therefore, increasing the wheel radius relative to the axle increases mechanical advantage.

Example:

Wheel radius = 30 cm
Axle radius = 3 cm$$MA=\frac{30}{3}=10$$

If the load is 1200 N:$$E=\frac{Load}{MA} =\frac{1200}{10}=120N$$

So a 120 N effort is sufficient for the idealised system.

3. Efficiency

Real machines experience friction, so they cannot normally convert all input work into useful output work.$$\eta= \frac{\text{Useful output work}} {\text{Total input work}}\times100\%$$

A real machine therefore has efficiency below 100%.

4. Gear trains

A gear train transfers motion and controls speed and torque.

The chapter uses a mechanical watch as an example, where different gears allow the seconds, minutes and hours hands to rotate at different speeds.

5. Tension

Tension is the pulling force transmitted through a stretched string, rope, thread or cable.

It:

• acts along the string;
• pulls away from the object;
• has SI unit newton.

For an object hanging at rest:$$T=mg$$

For a mass accelerating upward:$$T-mg=ma$$

Therefore:$$T=m(g+a)$$

For two unequal masses connected over a pulley, the heavier mass moves downward and the lighter mass upward. Their common string constraint gives them the same magnitude of acceleration.

Key idea

Machines do not create energy from nothing. They trade force, distance and/or speed to make useful tasks easier.

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Chapter 4 — The Geometry of Power: Advanced Simple Machines

Questions

A. Multiple Choice Questions

1. Mechanical advantage is the ratio of:
a) Effort to load
b) Load to effort
c) Work to time
d) Power to force

2. For an ideal wheel and axle:$$MA=$$

a) $\frac{R_a}{R_w}$
b) $\frac{R_w}{R_a}$
c) $R_w+R_a$
d) $R_w-R_a$

3. A larger wheel connected to a smaller axle generally provides:
a) Greater mechanical advantage
b) Zero mechanical advantage
c) Lower torque
d) No rotation

4. In an ideal machine, efficiency is:
a) 25%
b) 50%
c) 75%
d) 100%

5. Tension in a stretched rope acts:
a) Perpendicular to the rope
b) Along the rope
c) Only downward
d) Away from the pulley only

6. When equal weights hang on both sides of an ideal pulley, the system is:
a) Accelerated
b) Balanced
c) Falling
d) Rotating continuously

B. Fill in the Blanks

7. Mechanical advantage is the ratio of ______ to effort.

8. The SI unit of tension is ______.

9. A real machine has efficiency ______ than 100%.

10. A wheel and axle rotate ______.

11. When the forces on a body are balanced, it is in ______.

C. True or False

12. A machine can create energy from nothing.

13. Friction causes energy losses in real machines.

14. Tension acts along the length of a string.

15. Equal loads on the two sides of an ideal pulley produce a net unbalanced force.

D. Short Answer

16. What is mechanical advantage?

17. Why is the steering wheel of a vehicle larger than its steering axle?

18. Why is the efficiency of a real machine less than 100%?

19. What happens to a pulley system when one hanging mass is greater than the other?

E. Numerical Questions

20. A wheel has a radius of $30cm$ and its axle has a radius of $3cm$. Find the mechanical advantage.

21. If the load is $1200N$ and the mechanical advantage is 10, calculate the effort.

22. A $5kg$ mass hangs stationary from a rope. Take $g=9.8m/s^2$. Find the tension.

23. A $4kg$ mass accelerates upward at $2m/s^2$. Find the tension in the supporting rope.

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Answers

A. MCQ Answers

1. b) Load to effort
2. b) RwRa\frac{R_w}{R_a}
3. a) Greater mechanical advantage
4. d) 100%
5. b) Along the rope
6. b) Balanced

B. Fill in the Blanks

7. load
8. newton
9. less
10. together
11. equilibrium

C. True/False

12. False
13. True
14. True
15. False

D. Answers

16. Mechanical advantage is the ratio of load to effort:

$$MA=\frac{Load}{Effort}$$

17. A larger wheel radius produces greater torque for the same applied force, making it easier to turn the axle.
18. Some input energy is lost, particularly because of friction and other inefficiencies.
19. The system accelerates toward the side containing the greater mass.

E. Numerical Answers

$$MA=\frac{30}{3}$$$$\boxed{10}$$

$$E=\frac{Load}{MA}$$$$=\frac{1200}{10}$$$$\boxed{120N}$$

$$T=mg$$$$=5\times9.8$$$$\boxed{49N}$$

For upward acceleration,$$T=m(g+a)$$$$=4(9.8+2)$$$$\boxed{47.2N}$$

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