Chapter 4: The Geometry of Power – Advanced Simple Machines

Mechanical Advantage — The Force Multiplier

A machine can make work easier by multiplying force, changing the direction of an applied force, or changing speed and distance. It does not create energy: in an ideal machine, input work equals output work.

Mechanical Advantage (MA) = Load ÷ Effort

Efficiency (η) = (Useful output work ÷ Input work) × 100%

For a real machine, friction and other losses mean efficiency is generally less than 100%.

Activity: Why does a truck have a large steering wheel?

A larger wheel radius increases the turning effect (torque) produced by the driver's force. The wheel and axle arrangement helps the driver turn the steering column with less effort.

4.2 Wheel and Axle — The Steering Mastery

A wheel and axle consists of a large wheel fixed to a smaller axle. When effort is applied at the wheel rim, its larger radius produces a larger turning moment.

For an ideal wheel and axle:

MA = Radius of wheel ÷ Radius of axle

Example 1: A wheel radius is 40 cm and the axle radius is 5 cm. Find the ideal MA.

MA = 40/5 = 8.

Example 2: An ideal wheel and axle lifts an 800 N load. If MA = 8, find the effort.

MA = Load/Effort, so Effort = Load/MA = 800/8 = 100 N.

Quick Check: In a mechanical watch, the seconds-to-minutes and minutes-to-hours gear ratios are both 60:1. If the seconds gear is 2 mm, find the corresponding hour gear size.

Overall ratio = 60 × 60 = 3600:1.

Hour gear size = 2 mm × 3600 = 7200 mm = 7.2 m.

The fastest gear (seconds hand) is connected nearest the power source; the gear train reduces speed for the minute and hour hands.

4.3 Pulleys, Weight and Tension

A fixed pulley changes the direction of the applied force; by itself, an ideal fixed pulley does not multiply force. In an ideal, massless rope over a frictionless pulley, tension is the same throughout the rope.

Weight near Earth's surface: W = mg.

Question 1. Show the direction of weight and tension for two hanging objects.

Answer: Weight (mg) acts vertically downward on each object. Tension acts along the rope, away from each object; for a vertical hanging rope, it acts upward on each mass.

Question 2. An 8 kg mass hangs at rest from a fixed pulley. Find the tension.

At rest, T = mg = 8 × 9.8 = 78.4 N.

Question 3. Masses of 20 kg and 10 kg hang on opposite sides of an ideal pulley. Find the net downward force and direction.

Heavier mass = 20 kg.

Net force = (20 − 10)g = 10 × 9.8 = 98 N.

Answer: The 20 kg mass moves downward and the 10 kg mass moves upward; net driving force is 98 N.

Question 4. A 6 kg mass hangs at rest from a fixed pulley. Find the tension.

T = mg = 6 × 9.8 = 58.8 N.

Question 5. Two masses of 2 kg and 6 kg are connected over a frictionless pulley. Find acceleration and tension. Take g = 9.8 m/s².

The 6 kg mass moves down and the 2 kg mass moves up.

Acceleration: a = [(m₂ − m₁)g]/(m₁ + m₂)

a = [(6 − 2) × 9.8]/(6 + 2) = 39.2/8 = 4.9 m/s².

Tension using the lighter mass: T = m₁(g + a) = 2(9.8 + 4.9) = 29.4 N.

Additional example: A 5 kg object is stationary on a rope. Find tension.

For equilibrium, T = mg = 5 × 9.8 = 49 N.

Check Your Understanding — Answers

1. Why is the steering wheel larger than the steering column?

Answer: The larger radius produces greater torque for the same force, helping multiply the turning effect.

2. A machine has an ideal mechanical advantage of 10. What is its efficiency when friction is present?

Answer: It is less than 100%, because some input energy is dissipated, mainly as heat, due to friction. The value cannot be determined from ideal MA alone.

3. What happens to tension when a stationary hanging object is replaced by a heavier one?

Answer: Tension increases because for an object at rest T = mg, and the heavier object has greater weight.

4. A 5 kg mass hangs at rest. Take g = 9.8 m/s². Find the tension.

T = mg = 5 × 9.8 = 49 N.

5. A truck's steering wheel has radius 30 cm and its axle has radius 3 cm. Find the ideal mechanical advantage.

MA = 30/3 = 10.

6. Assertion: The hour-hand gear in a watch is larger than the seconds-hand gear. Reason: all watch hands must rotate at the same speed.

Answer: The assertion is generally true for the gear train described, but the reason is false. The hands must rotate at different speeds: the second hand fastest and the hour hand slowest.

7. Assertion: A heavier bob increases tension in a supporting thread. Reason: Tension acts along the thread and pulls on the attached object.

Answer: Both statements are true, but the reason is a definition of tension rather than the full explanation. For a stationary bob, tension rises because T = mg and the weight increases.

Formula Summary

• Mechanical advantage: MA = Load/Effort
• Ideal wheel-and-axle MA = R_wheel/R_axle
• Weight: W = mg
• Equilibrium for a single stationary hanging mass: T = mg
• Atwood-machine acceleration: a = [(m₂ − m₁)g]/(m₁ + m₂), for m₂ > m₁
• Atwood-machine tension: T = m₁(g + a) = m₂(g − a)
• Efficiency: η = (output work/input work) × 100%

Note: Pulley numerical answers assume an ideal, massless rope and frictionless pulley unless stated otherwise.