Page 288 - Physics Coursebook 2015 (A level)

Cambridge International A Level Physics
The gravitational potential at a point is the work done per unit mass in bringing a mass from infinity to the point.
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QUESTION
9 There is a point on the line joining the centres of the Earth and the Moon where their combined gravitational field strength is zero. Is this point closer to the Earth or to the Moon? Calculate how far it is from the centre of the Earth.
Energy in a gravitational field
As well as the force on a mass in a gravitational field, we can think about its energy. If you lift an object from the ground, you increase its gravitational potential energy (g.p.e.). The higher you lift it, the more work you do on it and so the greater its g.p.e. The object’s change in g.p.e. can be calculated as mgΔh, where Δh is the change in its height (as we saw in Chapter 5).
This approach is satisfactory when we are considering objects close to the Earth’s surface. However, we need a more general approach to calculating gravitational energy, for two reasons:
■■ If we use g.p.e. = mgΔh, we are assuming that an object’s g.p.e. is zero on the Earth’s surface. This is fine for
many practical purposes but not, for example, if we are considering objects moving through space, far from Earth. For these, there is nothing special about the Earth’s surface.
■■ If we lift an object to a great height, g decreases and
we would need to take this into account when calculating g.p.e.
For these reasons, we need to set up a different way of thinking about gravitational potential energy. We start by picturing a mass at infinity, that is, at an infinite distance from all other masses. We say that here the mass has zero potential energy. This is a more convenient way of defining the zero of g.p.e. than using the surface of the Earth.
Now we picture moving the mass to the point where we want to know its g.p.e. As with lifting an object from the ground, we determine the work done to move the mass to the point. The work done on it is equal to the energy transferred to it, i.e. its g.p.e., and that is how we can determine the g.p.e. of a particular mass.
Gravitational potential
In practice, it is more useful to talk about the gravitational potential at a point. This tells us the g.p.e. per unit mass at the point (just as field strength g tells us the force per unit mass at a point in a field). The symbol used for potential is φ (Greek letter phi), and unit mass means one kilogram. Gravitational potential is defined as follows:
For a point mass M, we can write an equation for φ at a distance r from M:
φ = − GrM
where G is the gravitational constant as before. Notice the minus sign; gravitational potential is always negative. This is because, as a mass is brought towards another mass, its g.p.e. decreases. Since g.p.e. is zero at infinity, it follows that, anywhere else, g.p.e. and potential are less than zero, i.e. they are negative.
Picture a spacecraft coming from a distant star to visit the solar system. The variation of the gravitational potential along its path is shown in Figure 18.8. We will concentrate on three parts of its journey:
1 As the craft approaches the Earth, it is attracted
towards it. The closer it gets to Earth, the lower its g.p.e.
becomes and so the lower its potential.
2 As it moves away from the Earth, it has to work against
the pull of the Earth’s gravity. Its g.p.e. increases and so we can say that the potential increases. The Earth’s gravitational field creates a giant ‘potential well’ in space. We live at the bottom of that well.
3 As it approaches the Sun, it is attracted into a much deeper well. The Sun’s mass is much greater than the Earth’s and so its pull is much stronger and the potential at its surface is more negative than on the Earth’s surface.
potential potential = 0 distance
Sun
Earth
Figure 18.8 The gravitational potential is zero at infinity (far from any mass), and decreases as a mass is approached.