The correct answer is A because:

The gravitational field strength from the star of mass M, g_M = $\frac{G M}{x^{2}}$
The gravitational field strength from the star of mass 4M, g_4M = $\frac{G (4 M)}{{(r - x)}^{2}} = \frac{4 G M}{{(r - x)}^{2}}$
This can be visualised using the diagram below:

7-1-mcq-q5-ma-hard-aqa-a-level-physics

If the resultant force is zero at some point on the straight line joining the centres of the stars, then the magnitude of the gravitational field strength is equal (and in opposite directions)
Therefore:
- $\frac{G M}{x^{2}} = \frac{4 G M}{{(r - x)}^{2}}$
- $\frac{1}{x^{2}} = \frac{4}{{(r - x)}^{2}}$
- (r– $x$ )² = 4 $x$ ²
Expanding the brackets gives:
- ${(r - x)}^{2}$ = r² – 2 $x$ r + $x$ ²
Therefore:
- r² – 2 $x$ r + $x$ ² = 4 $x$ ²
- r² – 2 $x$ r – 3 $x$ ² = 0
This is a quadratic equation in r, therefore we can use the quadratic formula to solve for r (in terms of $x$ )
- r = $\frac{- (- 2 x) \pm \sqrt{{(- 2 x)}^{2} - 4 (1) (- 3 x^{2})}}{2 (1)}$
- r = $\frac{2 x \pm \sqrt{4 x^{2} + 12 x^{2}}}{2}$
- r = $\frac{2 x \pm \sqrt{16 x^{2}}}{2}$
- r = $\frac{2 x \pm 4 x}{2}$
Therefore r = $\frac{6 x}{2}$ = 3 or r = – $\frac{2 x}{2}$ = – $x$
Because r and are distances, we can ignore the negative solution for r Therefore:
- r = 3 $x$
- $\frac{r}{x}$ = 3

As an able problem solver, you will be familiar with the various methods of solving equations with unknowns – e.g. setting up two simultaneous equations (if there are two unknowns) or using the quadratic formula (if there is a quadratic expression). The quadratic formula is on your data sheet, so don’t worry if you can’t memorise it.

The latter worked in this case, because we just wanted an equation of one in terms of the other, e.g. r in terms of $x$ . When attempting more challenging questions, where the way forward isn’t immediately clear, have a think about these mathematical tools and just have a go at using one of them; you will quickly determine whether it’s the right way forward, or not! As you gain experience with practise, you’ll get into the swing of spotting which approach best suits certain types of question.

Gravitational Fields (AQA A Level Physics)

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