Weight and gravitational field strength

Gravitational field strength, represented by $g$, is the gravitational force per kilogram of mass at a particular location. Its unit is newtons per kilogram (N/kg).

Near Earth’s surface, we usually use $g = 10\text{ N/kg}$ at GCSE. This means that each kilogram of mass experiences a gravitational force of approximately 10 N. A 2 kg object therefore has twice the weight of a 1 kg object in the same gravitational field.

Weight, mass and gravitational field strength are connected by:

$$W = m \times g$$

Here, $W$ is weight in newtons, $m$ is mass in kilograms and $g$ is gravitational field strength in newtons per kilogram. Multiplying kilograms by newtons per kilogram gives an answer in newtons.

You can also rearrange the equation to find mass or gravitational field strength:

$$m = \frac{W}{g} \qquad g = \frac{W}{m}$$

Dividing weight by gravitational field strength finds the mass experiencing that force. Dividing weight by mass finds the gravitational force on each kilogram.

Measuring weight with a newton meter

A newton meter, also called a spring balance, measures weight directly. It contains a spring, a hook for the object and a calibrated scale marked in newtons. The object pulls on the spring, stretching it; the scale converts this stretch into a force reading.

Choose a newton meter whose range can accommodate the object’s weight, and check that it reads zero before attaching the object. Hold or support the meter vertically and hang the object from its hook. Let the object hang freely without touching a surface. Once it is stationary, read the scale at eye level and record the weight in newtons.

A top-pan balance normally gives a reading of mass, rather than weight. You can measure mass with the balance and then calculate weight using $W = mg$, provided you know the gravitational field strength at that location. This is an indirect measurement of weight.

How weight changes with gravitational field strength

For an object whose mass stays constant, its weight is directly proportional to gravitational field strength. Doubling $g$ doubles its weight; halving $g$ halves its weight. The object has not gained or lost matter—the gravitational force acting on it has changed.

The Moon’s gravitational field strength is about $1.6\text{ N/kg}$, compared with about $10\text{ N/kg}$ on Earth. The same object therefore weighs roughly one-sixth as much on the Moon, while retaining the same mass. It is easier to support against gravity there because its weight is smaller.

The equation also shows a second relationship: at a fixed gravitational field strength, weight is directly proportional to mass. Twice as much mass experiences twice as much gravitational force.

$g$ is gravitational field strength in N/kg. Near Earth’s surface, use approximately $10\text{ N/kg}$ unless another value is given.

Measurement

• Direct: hang the object freely from a zeroed newton meter; read the stationary force in N.
• Indirect: measure mass with a balance, then calculate $W = mg$.

Relationships

• Fixed mass: $W \propto g$.
• Fixed gravitational field strength: $W \propto m$.
• Weaker gravitational field → smaller weight, but unchanged mass.