The lines in Figure 4 show the shape of the gravitational field around two stars G and H - AQA - A-Level Physics - Question 4 - 2022 - Paper 2

The field direction at point X is indicated by an arrow pointing away from G, while the field direction at point Y is shown by an arrow pointing toward H. This illustrates how the gravitational field behaves around the stars.

Using the formula for gravitational field strength at the surface of an asteroid, we have:

$g = \frac{GM}{R^2}$

Where:

• $g = 0.40 \, \text{N kg}^{-1}$ (gravitational field strength)
• $M = 2.0 \times 10^{20} \text{kg}$ (mass of P)
• $G = 6.67 \times 10^{-11} \text{N m}^2 \text{kg}^{-2}$ (gravitational constant)

Rearranging gives:

$R^2 = \frac{GM}{g}$

Substituting in the values:

$R^2 = \frac{(6.67 \times 10^{-11})(2.0 \times 10^{20})}{0.40}$

Calculating gives:

$R \approx 1.8 \times 10^{10} \, \text{m}$

The graph should depict a curve that starts at a higher value of g when r equals R, then decreases asymptotically as r increases. The value of g will be 0.40 N kg⁻¹ at r = R, and gradually approach zero as r increases towards 2R and 3R.

The area under the graph between r = R and r = 2R represents the work done against the gravitational field when moving from radius R to 2R. This work is a measure of the energy transferred per unit mass as the object moves further from the gravitational source.

To find the acceleration of asteroid P, we use Newton's second law:

$F = ma$

Where:

• $F = 6.38 \times 10^{12} \, \text{N}$ (gravitational force from G)
• $m = 2.0 \times 10^{20} \, \text{kg}$ (mass of P)

Rearranging gives:

$a = \frac{F}{m} = \frac{6.38 \times 10^{12}}{2.0 \times 10^{20}} = 3.19 \times 10^{-8} \, \text{m s}^{-2}$

Asteroid P cannot maintain a circular orbit around H due to the gravitational influence of both G and H. The combined gravitational pull from G and H would create a complex trajectory for P, making stable circular motion impossible. Furthermore, if the distance between P and H varies, P will experience changing gravitational forces, preventing it from achieving the necessary centripetal acceleration.