# Gravitational Field: Definition & Formula

## What Is a Gravitational Field?

A **gravitational field** is the force field that exists in the space around every mass or group of masses. This field extends out in all directions, but the magnitude of the gravitational force decreases as the distance from the object increases. It is measured in units of force per mass, usually newtons per kilogram (N/kg). A gravitational field is a type of force field and is analogous to electric and magnetic fields for electrically charged particles and magnets, respectively.

There are two ways of showing the gravitational field around an object: with arrows and with field lines. Both of these are shown in the picture below. Arrows show the magnitude and direction of the force at different points in space. The longer the arrow, the greater the magnitude. Field lines show the direction the force would act on an object placed at that point in space. The magnitude of the field is represented by the spacing of the lines. The closer the lines are to each other, the higher the magnitude.

The gravitational field varies slightly at the earth's surface. For example, the field is slightly stronger than average over subterranean lead deposits. Large caverns that may be filled with natural gas have a slightly weaker gravitational field. Geologists and prospectors of oil and minerals make precise measurements of the earth's gravitational field to predict what may be beneath the surface.

## Formula

The earth and moon exert a force, or pull, on each other even though they are not in contact. In other words, the two bodies interact with one another's gravitational field. Another example is the interaction of the earth and a satellite in orbit around it.

From these examples, Newton developed the **law of universal gravitation**. The law of universal gravitation says that every object exerts a gravitational pull on every other object. The force is proportional to the masses of both objects and inversely proportional to the square of the distance between them (or the distance between their centers of mass if they are spherical objects). Using variables, we write *F* is proportional to *mM*/*d*^2, where *F* is the force, *m* is the mass of the smaller object, *M* is the mass of the larger object, and *d* is the distance between the two objects.

In 1798, English physicist Henry Cavendish performed precise measurements of the actual gravitational forces acting between masses using a **torsion balance**. The outcome of his experiment resulted in the constant of proportionality in the law of universal gravitation called the **universal gravitational constant**. Inserting this into the proportionality results in the equation *F* = *G*(*mM*/*d*^2). The value for *G* is 6.67 x 10^-11 newton-meters squared per square kilogram (N-m^2/kg^2).

The following is a diagram of the torsion balance set up used by Cavendish to determine the universal gravitational constant.

When we talk about the gravitational force acting on an object, we can come up with an equation in a couple of different ways using the universal law of gravitation and **Newton's Second Law of Motion**. Newton's Second Law of Motion also describes force and says that an object is accelerated whenever a net external force acts on it. The net force equals the mass of the object times its acceleration. Mathematically, we have *F* = *ma*, where *F* is the force acting on the object, *m* is the mass of the object, and *a* is the acceleration of the object.

When a gravitational force acts on an object, we replace *a* with *g*. Solving for *g*, we have *g* = *F*/*m*. Gravitational field is measured in units of newtons/meter (N/m), which also reduces to meters per second squared (m/s^2). What the unit m/s^2 is really describing is the acceleration of the object, or the change in speed over time. Meters per second squared can also be expressed as meters per second per second (m/s/s). For example, the average value of the gravitational field on the surface of the earth is 9.8 m/s^2. This means that an object experiencing the earth's gravitational field (or free falling to the earth) has a speed of 9.8 m/s after the first second, after two seconds it is 19.6 m/s, after 3 seconds it is 29.4 m/s, and so on.

We can also come up with an alternative equation by setting the universal law of gravitation and Newton's Second Law of Motion equal to each other since they both describe force. Doing this, we have *mg* = *G*(*mM*/*d*^2). The mass of object 1 cancels out and we are left with *g* = *GM*/*d*^2.

## Lesson Summary

In this lesson, we learned that a **gravitational field** is the force field that exists in the space around every mass or group of masses. We also learned that gravitational fields are represented with field lines similar to electric and magnetic fields, but can also be represented by arrows with the length of the arrow indicating the field strength. We also presented two equations for the gravitational field, *g* = *F*/*m* and *g* = *GM*/*d*^2, based on the **universal law of gravitation** and **Newton's Second Law of Motion**. Gravitational field is measured in units of newtons per meter (N/m) or meters per second squared (m/s^2).

## Learning Outcomes

After reviewing this lesson, you should have the ability to:

- Define gravitational field
- Recall how gravitational fields are represented
- Identify the two equations for gravitational field and Newton's laws that these equations are based on
- Describe the units used to measure gravitational field

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