describe and use the concept of weight as the effect of a gravitational field on a mass and recall that the weight of an object is equal to the product of its mass and the acceleration of free fall

Momentum and Newton’s Laws of Motion

Learning Objective

Describe and use the concept of weight as the effect of a gravitational field on a mass, and recall that the weight of an object is equal to the product of its mass and the acceleration of free fall.

Key Concepts

• Mass (\$m\$) – a measure of the amount of matter in an object (unit: kilogram, kg).
• Weight (\$W\$) – the force exerted on a mass by a gravitational field (unit: newton, N).
• Acceleration due to gravity (\$g\$) – the rate at which objects accelerate towards the centre of the Earth; \$g \approx 9.81\ \text{m s}^{-2}\$ near the surface.
• Newton’s First Law – an object remains at rest or in uniform motion unless acted upon by a net external force.
• Newton’s Second Law – the net force on an object is equal to the rate of change of its momentum: \$\mathbf{F} = \dfrac{d\mathbf{p}}{dt}\$, and for constant mass, \$\mathbf{F}=m\mathbf{a}\$.
• Newton’s Third Law – for every action there is an equal and opposite reaction.

Weight as a Gravitational Force

The weight of an object is the gravitational force acting on its mass. It is given by the simple product:

\$\$

W = m g

\$\$

where:

• \$W\$ is the weight (N),
• \$m\$ is the mass (kg),
• \$g\$ is the local acceleration due to gravity (m s⁻²).

Direction of Weight

Weight always acts vertically downwards, towards the centre of the Earth. In vector form:

\$\$

\mathbf{W} = -\,m g\,\hat{\mathbf{j}}

\$\$

(\$\hat{\mathbf{j}}\$ denotes the upward unit vector; the negative sign indicates the downward direction.)

Weight vs. Mass – Common Misconceptions

1. Mass is an intrinsic property; weight depends on the gravitational field.
2. On the Moon, \$g_{\text{Moon}} \approx 1.62\ \text{m s}^{-2}\$, so an object’s weight is about one‑sixth of its Earth weight, but its mass remains unchanged.
3. Weight is a force and therefore measured in newtons, not kilograms.

Practical Example

Calculate the weight of a 75 kg student on Earth and on the Moon.

Relation to Momentum

Newton’s second law can be expressed in terms of momentum \$\mathbf{p}=m\mathbf{v}\$:

\$\$

\mathbf{F} = \frac{d\mathbf{p}}{dt}

\$\$

When the only external force is weight, the vertical component of momentum changes according to:

\$\$

\frac{dp_y}{dt} = -mg

\$\$

This links the concepts of weight (a force) and momentum (a product of mass and velocity).

Summary Checklist

• Weight is a force: \$W = mg\$.
• Mass is invariant; weight varies with \$g\$.
• Weight acts vertically downwards.
• Newton’s second law relates weight to the change in momentum.