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# Gravitational field strength

## Gravitational field definition

A gravitational field is a region in which a mass experiences a force due to gravitational attraction.

When an object is dropped from a height, it falls towards the surface of the earth. This is because there is gravitational attraction between the object and the earth.

Gravitational fields are caused by mass. All objects that have mass have a gravitational field around them. The strength of the field depends on how much mass the object has.

Humans, animals, plants and cars are all attracted to the Earth as it has lots of mass.

Two rocks do not noticeably attract each other as they do not have much mass.

Gravitational forces are always attractive. Two objects cannot repel each other because of gravity.

Water falls to towards the Earth because of the gravitational attraction.

## Advanced concepts in gravitational fields

The term gravitational field $\vecphy{g}$ has quite different meanings in informal and formal scientific settings.

• Informally, it can mean a region of space in which a gravitational force acts on a body. This is the meaning referred to when people say, for example, that there is a gravitational field around a star.

• In scientific settings, a gravitational field is a quantity that specifies the force (both the magnitude AND direction) acting on a unit mass at a specific point in space.

The value of a field function is different for different points in the field, due to the law of universal gravitation. This reflects, for example, that the force is weaker further away from a point mass.

The format of the function depends on the shape and alignment of masses.

The magnitude of the field at a point is equal to the gravitational field strength $g$. In other words, the gravitational field strength is the scalar of the gravitational field.

In principle, any mass will exert a gravitational force on an object placed anywhere in space. In practice as the distance increases the force becomes infinitesimally small even for large masses.

In a uniform field the gravitational field strength and the direction of the force is the same at any point in the field.

For small distances above the surface of the earth, the gravitational field can be treated as uniform. This is because the distance from the earth's surface to the atmosphere is very small compared to the radius of the earth.

## Gravitational field strength definition

The gravitational field strength ($g$) is the gravitational force per unit mass.

$g$ has units of $\text{N/kg}.$

At the surface of the Earth the gravitational field strength is roughly constant. Its value is $g = 9.81 \text{ N/kg} \approx 10 \text{ N/kg}.$

The gravitational field strength on other planets is different to on Earth. On Mars $g = 3.8 \text{ N/kg}$ and on Jupiter $g = 26 \text{ N/kg}.$

The gravitational field at the surface of the Sun is about 30 times larger than at the surface of the Earth.

## Gravitational field strength derivation

The gravitational field strength $(g)$ is a quantity that describes the gravitational effect of a mass on objects around it.

Specifically, $g$ is the magnitude of the gravitational force per unit mass (i.e. $1\text{ kg}$) at a point. $g$ is often casually referred to as the gravitational force on a unit mass.

This ignores the difference between a force (a vector) and the magnitude of the force (a scalar).

The equation for $g$ is derived from the equation for the law of universal gravitation:$$F=\frac{GMm}{r^2}$$Both sides of the equation are divided by $m$: $$g=\frac{F}{m}=\frac{GM}{r^2}$$

$F=$magnitude of force; $G=$universal gravitational constant; $M$ or $m=$mass; $r=$distance from a point mass.

## Gravitational potential energy

The gravitational potential energy ($E_{\text{p}}$ or $U$) of an object at a point is the work done by a gravitational field in bringing the object from infinity to the point.

$E_{\text{p}}$ itself depends on the object (its mass) and the point of its location (relative to the gravitational field around another object).

The equation for gravitational potential energy $E_{\text{p}}$ is: $$E_{\text{p}}=U =-\frac{GMm}{r}$$$G=$Universal gravitational constant; $m_1$ and $m_2=$two point masses; $r=$distance between the masses.

By convention, $E_{\text{p}}$ is set to be zero at infinity. At a finite distance $E_{\text{p}}$ is negative.

$E_{\text{p}}$ indicates the potential of the object to gain kinetic energy (to accelerate) simply because of its position in the gravitational field of another object.

A body accelerating because of a gravitational force reaches a higher velocity (and kinetic energy) if it starts at an infinite distance than if it starts from closer to the mass that is attracting it.

$E_{\text{p}}$ is often informally abbreviated as PE. The SI unit of gravitational potential energy is the joule ($\text{J}$).

## Acceleration due to gravity

The gravitational field strength $g$ is equivalent to the magnitude of acceleration due to gravity. This is the acceleration when an object is only subject to gravitational force (i.e. when it is in free fall).

This arises from Newton's second law:\begin{align*}&&F&=ma\\&\Rightarrow&\frac{GMm}{r^{2}}&=ma\\&\Rightarrow&\frac{GM}{r^{2}}&=a\\&\Rightarrow&g&=a\end{align*}

The phrase "gees of force" is often used colloquially to refer to the force which causes multiples of the magnitude of the gravitational acceleration on an object (i.e. $a=ng$, where n is an integer).

$g$ is measured in newtons per kilogram or metres per second squared: $1 \text{ N}/\text{kg}=1 \text{ m}/\text{s}^2$.