1.2 Motion – Weight as the Effect of a Gravitational Field
Learning Objective
Describe, and use, the concept of weight as the effect of a gravitational field on a mass.
Key Definitions
- Mass (m) – the amount of matter in an object. Scalar, invariant with location, measured in kilograms (kg).
- Weight (W) – the gravitational force exerted on a mass by a gravitational field. It is a vector, so it has both magnitude (in newtons, N) and direction (towards the centre of the attracting body).
- Gravitational field strength (g) – the force per unit mass in a gravitational field. On the Earth’s surface g ≈ 9.8 N kg⁻¹ (or 9.8 m s⁻²).
Relationship Between Mass, Weight and Gravitational Field
The weight of an object is directly proportional to its mass and to the local field strength:
\(W = m\,g\)
Re‑arranging gives the definition of field strength:
\(g = \dfrac{W}{m}\)
- W – weight (N)
- m – mass (kg)
- g – gravitational field strength (N kg⁻¹ or m s⁻²)
Units and Direction
- Weight: newton (1 N = 1 kg·m s⁻²).
- Gravitational field strength: N kg⁻¹ (equivalently m s⁻²).
- As a vector, weight always points towards the centre of the attracting body (e.g. downwards towards the centre of the Earth).
Mass vs. Weight – Comparison
| Property | Mass | Weight |
|---|
| Physical nature | Scalar (amount of matter) | Vector (force) |
| Symbol | \(m\) | \(W\) |
| SI unit | kilogram (kg) | newton (N) |
| Depends on location? | No | Yes – varies with g |
| Formula | – | \(W = m\,g\) |
Beam Balance – Comparing Masses, Not Weights
Effect of Changing Gravitational Field Strength
From \(W = m\,g\) we see that if g changes, the weight changes proportionally while the mass stays constant.
- Earth: \(g_{\text{Earth}} \approx 9.8\ \text{N kg}^{-1}\)
- Moon: \(g_{\text{Moon}} \approx 1.6\ \text{N kg}^{-1}\)
Worked Example
A 12 kg object is taken to the Moon.
- Calculate its weight on the Moon:
\(W{\text{Moon}} = m\,g{\text{Moon}} = 12\ \text{kg} \times 1.6\ \text{N kg}^{-1} = 19.2\ \text{N}\)
- For comparison, the same object on Earth weighs:
\(W_{\text{Earth}} = 12\ \text{kg} \times 9.8\ \text{N kg}^{-1} = 117.6\ \text{N}\)
- Notice that the mass (12 kg) is unchanged; only the weight varies with g.
Weightlessness (Extra – not a core requirement)
Weightlessness occurs when the net gravitational force on a body is zero.
- Deep space far from any massive body → \(g \approx 0\) → \(W = 0\).
- Free‑fall (e.g., an astronaut orbiting Earth). Both astronaut and spacecraft accelerate at the same rate g, so no contact force is felt and the sensation of weight disappears.
Measuring Weight
- Spring (elastic) scale – the extension \(x\) of a calibrated spring follows Hooke’s law \(F = kx\). The scale directly measures the force (weight) and then, by assuming a standard value of \(g\) (usually 9.8 m s⁻²), displays a “mass” reading in kg.
- Remember: a spring scale measures weight; the mass shown is obtained by dividing the measured force by the assumed \(g\).
Free‑Body Diagram (Illustrative)
Object of mass m resting on a horizontal surface:
- Weight vector \(W = mg\) points vertically downwards toward the centre of the Earth.
- Normal reaction R points vertically upwards and, in the absence of other vertical forces, has magnitude equal to W.
In a diagram draw an arrow labelled \(W\) from the centre of the object toward the ground, and an equal‑length arrow \(R\) upward.
Common Misconceptions
- “Weight is the same as mass.” – Weight depends on the gravitational field; mass does not.
- “If an object feels lighter, its mass has decreased.” – The mass is unchanged; either g is smaller or an upward force reduces the net weight.
- “A scale measures mass directly.” – Most classroom scales measure weight and then divide by an assumed \(g\) to give a mass reading.
- “A balance measures weight.” – A balance compares the *ratio* of forces, so it actually compares masses and works even if g changes.
Summary
Weight is the gravitational force acting on a mass, expressed by the vector equation \(W = mg\). It has magnitude (N) and direction (towards the centre of the attracting body) and varies with the local gravitational field strength, whereas mass is an invariant scalar. Mastery of the distinction, the ability to calculate weight in different environments, the use of a beam balance to compare masses, and an awareness of common misconceptions are essential for the Cambridge IGCSE Physics (0625) syllabus.