# Activity: Gravitational field

#### Aims

• Introduce the concept of a field.
• State Newton's universal law of gravitation.
• Define gravitational field strength.
• How field lines are used to represent a field.
• Derive the equation for a circular orbit.

#### Doing this at home

The subject matter in this topic can be readily experienced by anyone anywhere on the planet! And of course you can use simulations to take measurements and verify the laws.

#### Gravity

Gravity is what causes the force that attracts all objects with mass to the ground. The simulation below shows two objects close to the earth. Move them around and observe the force.

To describe a region of space where a body experiences a force we use the term field. A gravitational field is a region where a mass experiences a force due to its mass. The force is dependent on the mass of the body F = mg

#### Field strength (g)

The force on an object depends on its mass but the force per unit mass is the same, this quantity is used to define the strength of the field.

The field strength is the force per unit mass experienced by a small test mass placed in the field.

• What is the gravitational force experienced by a 1kg mass placed close to the Earth?
• What is the field strength on the surface of the Earth?

#### Field lines

Field lines are drawn to show the direction and strength of the field. The density of lines gives the strength and their direction the direction.

• Which field lines represent the strongest field?

If we move our test mass a long way from the earth the force would become less.

• Sketch the field lines for this situation.

In this case the field strength is proportional to 1/r2

• What would the field strength be at a distance of 6400 km form the surface of the Earth? (Earth radius = 6400 km)

Note that the simulation doesn't give the correct result inside the Earth. What should happen?

#### Newtons Universal Law of Gravitation

Newton wasn't able to go into space to measure the gravitational force but realised that the origin of the force causing an apple to fall to the earth was the same as the force holding the planets in orbit around the Sun. He was also convinced that this could be explained with a single mathematical equation.

Every particle of mass attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to their separation squared.

G is the universal gravitional constant

This was quite a bold statement since, at the time, the force between masses on the Earth was too small to measure.

Note that this simulation uses spheres of mass not points. Newton developed calculus to prove that a sphere of mass acts as if all the mass was at the centre.

• Use the Gravity Force Lab to find the Universal Gravitational constant G.
• What are the units of G?

G = 6.6742 x 10-11 m3kg-1s-2

• Why is the force on each ball the same even when they are a different size?
• Show that the field strength on the surface of the earth is approximately 10 Nkg-1. (Mass of Earth = 6 x 1024 kg)

Field strength is a vector quantity so must be added vectorially. Using the simulation below you can observe how the resultant field changes as you move around the field

You can use the simulation to check your solutions to the following problems.

• Find the resultant field in the positions shown.

Find the position where the field is zero.

#### Orbits

The gravitational force can provide the centripetal force required to cause a body to move in a circle. Consider a mass m orbiting a larger mass M as shown below.

• Write an expression for the Gravitational force experienced by m.
• Write an expression for the centripetal force needed to keep m in orbit.
• Equate these two expressions and show that

You can investigate orbits further in Algodoo Orbit simulation (Algodoo)

Orbits problems