2023 A/L Physics Past Paper · Question 02 (English Medium)

Answer: (2) $-\dfrac{GMm}{r}$

What the question asks

We want: the gravitational potential energy of the two spheres.

What each letter means:

$G$ ($\text{N m}^2\ \text{kg}^{-2}$): the gravitational constant.

$M$ ($\text{kg}$): the mass of the first sphere.

$m$ ($\text{kg}$): the mass of the second sphere.

$r$ ($\text{m}$): the distance between the centres of the spheres.

$E$ ($\text{J}$): the gravitational potential energy of the two masses.

Relevant theory

Since any mass placed at a certain point in a gravitational field is subjected to a gravitational force, a certain amount of work is done by the field in bringing this mass from outside the field (infinity) to this point.

Assuming the whole mass of the sphere is accumulated at its centre.

Gravitational potential at point A $= -G\dfrac{M}{R}$

Potential energy possessed by the body at point A $= -G\dfrac{Mm}{R}$

The negative sign above indicates that the relevant work is done by the field and not against the field.

The gravitational force of attraction between two particles of masses M and m situated at a distance r is given by,

Reading the options

What is happening:

The two spheres pull each other. The pull gets bigger when they are nearer, and it does work as they come together.

For example, think of a ball rolling down into a deep hole.

Far from the hole the ground is the zero level. As the ball goes down into the hole, the pull of the Earth does the work. The ball ends up below the zero level.

The two spheres are in the same way. The energy is measured from the level where they are very far apart. As they come closer, the field does the work, and the energy goes below zero.

So the energy of the two spheres is a negative number.

The first quote says that the energy is measured by the work the field does in bringing a mass in from outside the field.

The second quote says that each sphere can be taken as if all its mass were at its centre. That is why the distance is the one between the centres.

The third and fourth quotes give, for a body at the distance $R$ from the centre of the Earth, first the potential, which is the energy of one kilogram, and then the energy of the body.

$$-G\dfrac{M}{R} \quad\text{and}\quad -G\dfrac{Mm}{R}$$

In words: the energy of the mass $m$ is $m$ times the potential.

Here the distance between the centres is $r$, so we write $r$ in place of $R$.

$$E = -\dfrac{GMm}{r}$$

Check the units.

For $\dfrac{GMm}{r}$ we get $\text{N m}^2\ \text{kg}^{-2} \times \text{kg} \times \text{kg} \div \text{m} = \text{N m} = \text{J}$. This is an energy.

For $\dfrac{GMm}{r^2}$ we get $\text{N}$. This is a force, the pull in the last quote.

For $\dfrac{GM}{r}$ we get $\text{N m kg}^{-1} = \text{J kg}^{-1}$. This is an energy for each kilogram, the potential.

Test each option.

(1) $\dfrac{GMm}{r}$ has the right size, but a plus sign.

(2) $-\dfrac{GMm}{r}$ has the minus sign, the $m$, and $r$ once underneath. It matches.

(3) $\dfrac{GMm}{r^2}$ is the pull, in $\text{N}$.

(4) $-\dfrac{GMm}{r^2}$ is in $\text{N}$ too.

(5) $-\dfrac{GM}{r}$ has no $m$, so it is the potential.

Only option (2) matches.

Why the other options fail

The size is right but the sign is wrong. The minus sign shows that the work is done by the field.

$\dfrac{GMm}{r^2}$ is in $\text{N}$. It is the pull between the two masses, not an energy.

$-\dfrac{GMm}{r^2}$ is in $\text{N}$, so it is a force and not an energy.

$-\dfrac{GM}{r}$ has no $m$. It is the potential, the energy of one kilogram.

Conclusion

$E = -\dfrac{GMm}{r}$

The energy of the two spheres is $-\dfrac{GMm}{r}$.

The answer is (2).

For two masses, count the powers of $r$. The pull has $r^2$ underneath, and the energy has $r$ underneath.

A formula with no $m$ is the potential, the energy of one kilogram.

Exam tip