A-level Physics (Advancing Physics)/Gravitational Forces

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Gravity is a force. Any object with mass exerts a gravitational force on any other object with mass. The attractive force of one object on another is proportional to the product of their masses. However, this force is also inversely proportional to the distance between the objects squared. This means that, if two objects are twice as far away, the forces they exert on each other are four times smaller.

Thus gravitational force exerted by a sphere of mass M on another sphere of mass m is given by the following formula:

F_{grav} = \frac{GMm}{r^2},

where r is the distance between the spheres, and G is the Gravitational constant. Experiments have shown that G = 6.67 x 10-11 Nm2kg-2.

Gravitational Force Inside an Object[edit]

A lift acting under gravity in a lift shaft going through the centre of the Earth.

Inside a roughly spherical object (such as the Earth), it can be proved geometrically that the effects of the gravitational force resulting from all the mass outside a radius at which an object is located can be ignored, since it all cancels itself out. So, the only mass we need to consider is the mass inside the radius at which the object is located. The density of an object ρ is given by the following equation:

\rho = \frac{M}{V},

where M is mass, and V is volume. Therefore:

M = \rho V

If we substitute the volume of a sphere for V:

M = \frac{4}{3}\pi\rho r^3

And if we substitute this mass into the formula for gravitational force given above:

F_{grav} = \frac{-Gm\frac{4}{3}\pi\rho r^3}{r^2} = -\frac{4}{3}\pi G\rho mr

In other words, inside a sphere of uniform mass, the gravitational force is directly proportional to the distance of an object from the centre of the sphere. Incidentally, this results in a simple harmonic oscillator such as the one on the right. This means that a graph of gravitational force against distance from the centre of a sphere with uniform density looks like this:

Grav force sphere.svg

Questions[edit]

1. Jupiter orbits the Sun at a radius of around 7.8 x 1011m. The mass of Jupiter is 1.9 x 1027kg, and the mass of the Sun is 2.0 x 1030kg. What is the gravitational force acting on Jupiter? What is the gravitational force acting on the Sun?

2. The force exerted by the Sun on an object at a certain distance is 106N. The object travels half the distance to the Sun. What is the force exerted by the Sun on the object now?

3. What is the magnitude of the gravitational force that two 1kg weights exert on each other when they are 5cm apart?

4. The radius of the Earth is 6360km, and its mass is 5.97 x 1024kg. What is the difference between the gravitational force on 1kg at the top of your body, and on 1kg at your head, and 1kg at your feet? (Assume that you are 2m tall.)

Worked Solutions