Unknown mass using the principle of moments

Practical method

Aim

Use the principle of moments to determine an unknown mass and the mass of a uniform rule.

Variables

Independent variable: position of the known mass
Dependent variable: balance position (distance from the pivot)
Control variables: the same rule and pivot; masses hung vertically; distances measured from the pivot.

Apparatus

• Metre rule
• Knife-edge pivot
• 100 g slotted mass
• Unknown mass
• Thread loops
• Balance for checking

Method

1. Find the balance point of the uniform rule alone; this is its centre of mass (close to 50.0 cm).
2. Unknown mass: balance the rule at its centre of mass, hang the unknown mass at a fixed distance on one side and slide the 100 g mass until the rule balances. Record both distances and use the principle of moments.
3. Mass of the uniform rule: move the pivot to the 40.0 cm mark, hang the 100 g mass on the short side and slide it until the rule balances.
4. The rule's weight acts at its centre of mass, so clockwise and anticlockwise moments about the pivot give the rule's mass.
5. Repeat at several pivot positions and average.

Sample results

Uniform rule, centre of mass found at 50.0 cm; 100 g mass on the short side of the pivot.

Analysis

In each run the anticlockwise moment of the 100 g mass equals the clockwise moment of the rule's weight acting at its centre of mass: 0.100 × d₁ = M × d₂. All three pivot positions give M = 0.300 kg.

Uncertainty

Distances are read to ±0.1 cm, but judging the exact balance position adds about ±0.3 cm. Over a 10.0 cm lever arm this is a 3% uncertainty, the largest contribution.

Errors and improvements

Risk assessment

Common mistakes

• Measuring distances from the end of the rule instead of the pivot.
• Ignoring the rule's own weight when the pivot is not at its centre.
• Mixing grams and kilograms.

Worked analysis

Pivot at 40.0 cm; the 100 g mass at 10.0 cm balances the rule. Anticlockwise moment = 0.100 × 9.81 × 0.300 = 0.294 N m. Clockwise moment = M × 9.81 × 0.100. Equating gives M = \(\dfrac{0.294}{0.981}\) = 0.300 kg.

Common misconception

The weight of a uniform rule acts at its centre of mass, so it produces a moment whenever the pivot is not at the centre.

Variations required by some exam boards

Uniform rule (CCEA): with the pivot away from the centre of mass, the rule's own weight provides one of the moments, so its mass can be found with a single known mass.