Know that weights (and masses) may be compared using a balance

1.2 Motion – Comparing Weights and Masses Using a Balance

Learning Objectives (Cambridge AO1‑AO3)

• Explain how a balance can be used to compare the weights (and therefore the masses) of different objects.
• Distinguish clearly between scalar (mass, speed, distance) and vector (weight, velocity, force) quantities.
• Apply basic measurement techniques – reading, significant figures, uncertainty, averaging – in all IGCSE Physics contexts.
• Link the static equilibrium of a balance to Newton’s 1st law and to the concept of apparent weight when the system is accelerated.
• Use the fundamental motion equations  \(v=s/t\)  and \(a=\Delta v/\Delta t\) to solve simple problems.
• Combine forces vectorially and understand the role of torque in equilibrium.

Syllabus Connections

• 1.1 Physical quantities & measurement techniques
• 1.2 Motion – vectors, resultant forces, Newton’s 1st law
• 1.3 Mass, weight and density
• 1.4 Energy, work and power (via the work‑energy link to forces)

Measurement Toolbox (Syllabus 1.1)

All candidates must be familiar with the standard laboratory instruments used throughout the IGCSE Physics syllabus.

Simple Averaging Activity (AO1 – Data handling)

1. Measure the period of a simple pendulum 5 times using a stopwatch.
2. Record the values (e.g. 1.02 s, 1.04 s, 1.03 s, 1.01 s, 1.03 s).
3. Calculate the mean: \(\displaystyle \bar T = \frac{\sum T_i}{5}=1.03\ \text{s}\).
4. Quote the result with the appropriate number of significant figures and an uncertainty (e.g. \(1.03\pm0.02\ \text{s}\)).

Key Physical Quantities

• Mass (m) – amount of matter; scalar; SI unit kg (or g).
• Weight (W) – gravitational force on a mass; vector directed toward the centre of the Earth; SI unit N. \(W = mg\) (with \(g\approx9.8\ \text{m s}^{-2}\) on Earth).
• Force (F) – any interaction that can change the state of motion; vector; SI unit N.
• Speed (v) – scalar rate of change of distance; \(v = s/t\); unit m s\(^{-1}\).
• Velocity (→v) – vector rate of change of displacement; direction matters.
• Acceleration (a) – vector rate of change of velocity; \(a = \Delta v/\Delta t\); unit m s\(^{-2}\).

Scalars vs. Vectors (quick recap)

Why Use a Balance?

A balance does not measure weight directly. It compares the gravitational forces acting on two masses placed at equal distances from a fulcrum. Because the local acceleration due to gravity \(g\) is the same for both sides, the ratio of the forces equals the ratio of the masses. Hence a balance provides a reliable way to compare masses, and, by using \(W=mg\), also to compare weights without needing to know the numerical value of \(g\).

Principle of Operation

1. Place the unknown object on one pan.
2. Place known standard masses on the opposite pan.
3. Adjust the standards until the beam is horizontal (static equilibrium).
4. When equilibrium is achieved, the total mass on each side is equal:

   \[\displaystyle m{\text{unknown}} = \sum m{\text{standard}}\]

5. If required, calculate the weight: \(W = mg\).

Step‑by‑Step Procedure (including uncertainty)

1. Ensure the balance sits on a level surface; use the built‑in spirit level or a separate level.
2. Zero (tare) the balance if it has a tare function.
3. Gently place the unknown object on the left pan.
4. Gradually add standard masses to the right pan, noting each addition.
5. When the beam is horizontal, record the total mass of the standards.

   
   Uncertainty: add the balance’s typical reading uncertainty (±0.5 g) to the final result.

6. Express the mass to the correct number of significant figures (usually that of the least‑precise standard used).
7. Calculate the weight, if required, using \(W = mg\) and keep the same number of significant figures.

Worked Example – Determining an Unknown Mass

Standard masses available: 50 g, 20 g, 10 g, 5 g.

1. Place the metal block on the left pan.
2. Add a 50 g mass to the right pan – the left side still drops.
3. Add a 20 g mass – the left side still drops.
4. Add a 10 g mass – the beam becomes horizontal.

Result:

• Mass of block = \(50\ \text{g}+20\ \text{g}+10\ \text{g}=80\ \text{g}\) (±0.5 g).
• Weight on Earth = \(W = (0.080\ \text{kg})(9.8\ \text{m s}^{-2}) = 0.784\ \text{N}\) → \(0.78\ \text{N}\) (2 sf).

Comparison Table: Mass vs. Weight

Common Misconceptions

• “A balance measures weight.” It actually compares forces; because \(g\) is the same on both sides, the comparison is equivalent to comparing masses.
• “Objects with the same mass but different densities will tip the balance.” Density is irrelevant; only total mass matters.
• “Levelness is unimportant.” An unlevel balance changes the effective lever arms, introducing a systematic error.
• “Weight is a scalar.” Weight is a vector; its direction is always vertically downwards (or opposite to the normal reaction).

Link to Motion (Syllabus 1.2)

The balance is a concrete illustration of static equilibrium: the net force and net torque on the beam are zero. This is the stationary counterpart of Newton’s 1st law (“an object at rest stays at rest unless acted on by a net force”).

Static Equilibrium Condition

\[

\sum \tau = 0 \qquad\text{and}\qquad \sum F = 0

\]

where \(\tau = Fr\) is the torque about the fulcrum. Because the arms are equal, the condition reduces to \(F{\text{left}} = F{\text{right}}\), i.e. \(m{\text{left}} = m{\text{right}}\).

Motion Basics (Syllabus 1.1 & 1.2)

• Distance (s) – scalar, measured in metres (m).
• Displacement (→s) – vector, includes direction.
• Speed (v) – scalar, \(v = s/t\).
• Velocity (→v) – vector, \(→v = →s/t\).
• Acceleration (a) – vector, \(a = \Delta →v/\Delta t\).
• Key equations for uniformly accelerated motion (relevant for the extension activity):

  \[

  v = u + at,\qquad

  s = ut + \tfrac12 at^{2},\qquad

  v^{2}=u^{2}+2as

  \]

  where \(u\) is the initial speed.

Resultant Forces and Moments

When two or more forces act on a body, they can be combined into a single resultant using vector addition. For forces acting at right angles:

\[

F{\text{R}} = \sqrt{F{1}^{2}+F_{2}^{2}}

\]

Torque (moment) about a pivot is:

\[

\tau = F\,r\,\sin\theta

\]

In a balance the lever arms are equal and \(\theta = 0^{\circ}\) (forces act vertically), so \(\tau = Fr\) and equilibrium requires equal torques.

Example – Resultant of Perpendicular Forces

A 5 N force to the right and a 12 N force upward act on a block. The resultant magnitude is \(\sqrt{5^{2}+12^{2}} = 13\ \text{N}\) directed \(\tan^{-1}(12/5) \approx 67^{\circ}\) above the horizontal.

Extension Activity – Apparent Weight on an Accelerating Platform

1. Mount a small beam balance on a low‑friction air track so it can move horizontally without significant friction.
2. Place a known mass on the left pan and standard masses on the right pan, achieving equilibrium at rest.
3. Accelerate the track uniformly (e.g., using a motorised cart). Observe the tilt of the beam.
4. Explain the observation with the apparent‑weight formula:

   \[

   W_{\text{apparent}} = m(g + a)\quad\text{(upward acceleration)}

   \]

   \[

   W_{\text{apparent}} = m(g - a)\quad\text{(downward acceleration)}

   \]

5. Discuss the special case of free‑fall (\(a = -g\)) where the apparent weight becomes zero.

Summary

• A balance compares the gravitational forces on two masses; because \(g\) is the same for both, the comparison directly yields the ratio of the masses.
• Mass is a scalar; weight is a vector given by \(W = mg\) and varies with the local value of \(g\).
• Accurate use of a balance requires a level surface, awareness of the instrument’s uncertainty (±0.5 g), and correct handling of significant figures.
• The principle of static equilibrium illustrated by the balance underpins Newton’s 1st law and provides a bridge to motion concepts such as apparent weight, acceleration, and torque.
• Understanding vectors, resultant forces and basic motion equations completes the IGCSE requirement for the whole “Motion, forces & energy” block.

Practice Questions

1. A student balances an unknown object with a 200 g standard mass and a 50 g standard mass on the opposite pan. State the mass of the unknown object with its uncertainty and calculate its weight on Earth.
2. Explain why a balance would give the same reading for an object on Earth and the same object on the Moon, even though the actual weight is different.
3. Two objects have identical masses but different shapes (e.g., a solid sphere and a hollow cylinder). Will a balance show any difference when they are compared? Justify your answer.
4. During the extension activity, a 100 g mass on the left pan appears to weigh 120 g when the platform accelerates upward at \(2\ \text{m s}^{-2}\). Verify this observation using the apparent‑weight formula.
5. Measure the period of a pendulum five times (1.02 s, 1.04 s, 1.03 s, 1.01 s, 1.03 s). Calculate the mean period and express the result with the appropriate uncertainty and significant figures.
6. A car travels 30 m in 5 s with a constant speed. Calculate its speed and state whether speed is a scalar or a vector quantity.
7. A force of 8 N acts east and a force of 6 N acts north on a crate. Determine the magnitude and direction of the resultant force.
8. A 0.5 kg block is pulled down a frictionless incline of 30° by a horizontal force of 5 N. Resolve the forces parallel and perpendicular to the incline and find the block’s acceleration down the slope.