6: Fields and Forces

6.1 Gravitational Force and Field

Gravitational Fields

Mass and Weight• Mass is a measure of how much matter is in something.• Weight is the gravitational force acting upon an object.

Fields• A field is a region in which an object may experience a force. E.g. Charged bodies in electric fields.

Gravitational Fields

• Any mass has an associated gravitational field in which other masses will experience a force.

• The bigger the masses involved, the greater the gravitational forces between them.

Mass

Newton’s Universal Law of Gravitation

This states that if all masses are treated as point masses…

“All masses exert a gravitational force on all other masses. This force is proportional to the product of the masses and inversely proportional to the square of their separation.”

m1m2

F1 F2 F1 = F2 = Gm1m2

r2

This leads to…

Where G = The universal gravitational constant

= 6.67 x 10-11 Nm2kg-2

E.g.

Earth has a mass of 5.88 x 1024kg and a radius of 6.37 x 106m. Find the gravitational force on a 1.00kg mass at

a. sea level

b. the top of Monte Cervino (4478m)

Gravitational Field Strength

The field strength due to a large mass M is defined as…

so… but

so…

…the force per unit mass experienced by a small test mass (m) placed in the field.

F = GMm r2

g = F m

g = GM r2

g = gravitational field strength (Nkg-1)

This formula allows us to calculate g at any height above a large body of mass M (where height = r – radius of M).

E.g.

Mars has a mass of 6.42 x 1023kg and radius (r) 3397km. Calculate the gravitational field strength at its surface and at a point 2r from its centre.

g = 3.7 Nkg-1

g = 0.9 Nkg-1

Clearly when r doubles, g decreases by four. This is an inverse square relationship.

Sketch a graph of g against distance from centre of Earth:

g /

Nkg

-1

r / Earth radii

10.0

2.5

5.0

7.5

0

0 re 3re2re 4re

M

Field Lines

Field lines show the direction in which a point mass would accelerate.

• As with all field line diagrams, the strength of the gravitational field can be represented by how far apart the field lines are.

• Within any small region of a large Mass’ field, the field lines are virtually parallel and so the field can be considered uniform.

E.g. on Earth’s surface:

M• As distance from Earth increases, g becomes less.

Addition of fields

The resultant field from two or more masses can be found by vector addition.

E.g.

How would you determine the resultant fields at points A, B and C?

Earth MoonBA

C

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