2022 A/L Physics Past Paper · Question 49 (English Medium)

Answer: (3) Graph (3)

What the question asks

What we know:

A spacecraft goes in a straight line from the Earth to the Moon. $s$ is how far it has gone.

We neglect the Sun and the planets, so only the Earth and the Moon pull on the spacecraft.

There are no numbers, so we keep the letters. Later we check with sample numbers.

We want: the graph of the size of the net weight $W$ against the distance $s$.

What each letter means:

$W$ ($\text{N}$): the size of the net weight of the spacecraft, the pull of the Earth and the Moon together.

$s$ ($\text{m}$): the distance the spacecraft has gone from the Earth.

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

$M_E$, $M_M$ ($\text{kg}$): the mass of the Earth and the mass of the Moon.

$r$ ($\text{m}$): the distance of the spacecraft from the centre of the Earth.

$d$ ($\text{m}$): the distance of the spacecraft from the centre of the Moon.

Relevant theory

What is happening:

The Earth pulls the spacecraft back towards the Earth. The Moon pulls it on towards the Moon. The two pulls are in opposite directions, so the net weight is their difference.

At the start the spacecraft is close to the Earth. The Earth's pull is big and the Moon's pull is tiny. $W$ is big.

As the spacecraft goes on, the Earth's pull gets smaller and the Moon's pull gets bigger. $W$ falls.

At one place the two pulls are equal. The net weight is zero there.

Past that place the Moon pulls harder. The net weight points to the Moon and grows again until the spacecraft reaches the Moon.

The Moon is much lighter than the Earth, so the zero place is much closer to the Moon than to the Earth.

That force is called the weight of the body.

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

The strength of a gravitational field at any point in it is measured by the quantity known as the " Gravitational field intensity " at that point. The intensity of a gravitational field at such a point is the force acting on a unit mass placed at that point.

Hence the above intensity is a vector quantity directed towards the mass m.

The first quote says that the weight of a body is the resultant force of gravity on it. Here the resultant is the Earth's pull and the Moon's pull together.

The second quote is the law for the pull between two masses, and it depends on the distance $r$ between them. The pull gets smaller as the distance gets bigger.

The third and fourth quotes say that the pull on a unit mass at a point is towards the mass that causes it. So the Earth's pull is towards the Earth and the Moon's pull is towards the Moon.

For example, walking from a big loudspeaker towards a small one. The big one gets quieter and the small one gets louder. There is a spot, much nearer the small speaker, where they sound equally loud. Past that spot the small speaker is the louder one.

Required equations

$F_E = G\dfrac{M_E m}{r^2}$

$F_M = G\dfrac{M_M m}{d^2}$

$W = F_E - F_M$

$F_E = F_M$

Reasoning and calculation

Both pulls are forces in $\text{N}$, so their difference $W$ is in $\text{N}$.

$r$ and $d$ are both distances. As $s$ grows, $r$ grows and $d$ shrinks. That is why $F_E$ falls and $F_M$ rises.

$3.2$

$W = \dfrac{81}{5^2} - \dfrac{1}{5^2}$

$0$

$W = \dfrac{81}{9^2} - \dfrac{1}{1^2}$

$3.1$

$\left|\dfrac{81}{9.5^2} - \dfrac{1}{0.5^2}\right|$

With these sample numbers $W$ is $3.2$ halfway, $0$ at $r = 9$, and then $3.1$ at $r = 9.5$. So $W$ falls to zero near the Moon and then rises again.

Check the five graphs.

Graph (1) has its zero about a third of the way. That is too early, because the Moon is much lighter than the Earth.

Graphs (2) and (4) do not rise at the end. They leave out the Moon's pull.

Graph (5) never reaches zero, but there is a place where the two pulls are equal.

Graph (3) is the only one that falls to zero near the Moon and then rises. It is the best of the five.

The graphs are sketches, so we compare their features: where the zero is, and whether $W$ rises again.

Checks

$81 / (r * r) - 1 / ((10 - r) * (10 - r))$

Does it make sense?

At the start $W$ is the full weight of the spacecraft at the Earth. At the Moon $W$ is the weight of the spacecraft on the Moon, which is smaller than its weight on the Earth. So the graph should end lower than it started, and graph (3) does that.

The size of the net weight is never negative, so a graph never goes below the horizontal axis. Past the zero place the net weight points the other way, and we draw only its size, so the graph rises.

Conclusion

$\text{graph }(3)$

$W$ falls to zero near the Moon and then rises. The graph is (3).

The answer is (3).

Between two bodies the net pull is the Earth's pull minus the Moon's pull.

Look for the place where the two are equal. The net weight is zero there, and it grows again past it. The place is nearer the lighter body.

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