6.1 - Gravitational Force& Fields

Newton’s Universal Law of GravitationEvery single point mass attracts every other point mass with a force directly proportional to the product of the masses and inversely proportional to the square of their separation.

𝐹∝𝑚1𝑚2𝐹∝

1

𝑟 2

Universal Constant of Gravitation: G= 6.6742 x 10-11 m3kg-1s-2

Newton’s Universal Law of GravitationNewton calculated (using calculus) that Spheres also follow the same rule as long as the separation is between their centre of mass.

The Centre of Mass is a point that represents the total mass of a body and is where gravity can be said to act. For regular shapes it’s in the middle.

r

http://phet.colorado.edu/en/simulation/gravity-force-lab or click on the picture

Newton’s Universal Law of Gravitation on PhET

Use this to check Newton’s Law of Gravitation. What happens to the force if you double the distance? or double either of the masses?

Acceleration due to Gravity

Newton’s 2nd Law

𝐹=𝑚𝑎+ m

M

a

𝑎=𝐺𝑀𝑟 2

Newton’s Law of Gravitation

𝐹=𝐺𝑀𝑚𝑟2

On Earth: a = 9.81ms-2 Newton was right. This is correct!

Orbits: Centripetal Force & Gravity

𝐹=𝐺𝑀𝑚𝑟2

𝐹=𝑚𝑣2

𝑟

Assuming the orbits are circular.

Gravity causes the centripetal force

𝐺 𝑀𝑚𝑟2

=𝑚𝑣2

𝑟

𝒎𝟏

𝒎𝟐

+

𝑣=√𝐺𝑀𝑟

This is the speed of the orbiting object

Orbits: Centripetal Force & Gravity (cont)

𝑣=√𝐺𝑀𝑟

1. The speed of the earth in orbit around the sun:

m1 = mass of the sun = 1.99 x 1030 kgr = distance between Sun and Earth = 1.49 x 1011mG = 6.6742 x 10-11 m3kg-1s-2

= 29 846 ms-1

2. The time for the earth to orbit around the sun:

𝑣=𝑠𝑡

𝑠=2𝜋 𝑟𝑡=2𝜋 𝑟𝑣 𝑡=3.14×107𝑠

Newton was right again – 1 Year

Orbits: Centripetal Force & Gravity

http://phet.colorado.edu/sims/my-solar-system/my-solar-system_en.html

My Solar SystemGravity and Orbits

http://phet.colorado.edu/en/simulation/gravity-and-orbits

or click on the pictures

Try out these two great simulations to understand gravity even more.

Gravitational Field StrengthDefinition: The force per unit mass experienced by a small test mass* placed in the field.

units: Nkg-1

𝐹=𝐺𝑀𝑚𝑟2

𝑔=𝐺𝑀𝑟 2

For the mass m:

* A small test mass is used because any larger mass might change the field.

Field LinesThese show the direction that a mass would accelerate if placed in the field, and help us to imagine the field.

Around a spherical mass the field lines are closer together nearer the surface, so the field strength is larger.

Near the Earth the field lines are almost parallel.

The field is uniform.

Wherever you are near the surface of the earth you are pulled down with the same Force/Kilogram

Field Strength is a vector, so two values of g can be added together