A-level Physics (Advancing Physics)/Gravitational Forces/Worked Solutions

1. Jupiter orbits the Sun at a radius of around 7.8 x 10¹¹m. The mass of Jupiter is 1.9 x 10²⁷kg, and the mass of the Sun is 2.0 x 10³⁰kg. What is the gravitational force acting on Jupiter? What is the gravitational force acting on the Sun?

$F_{grav}=\frac{-GMm}{r^2}=\frac{-6.67\times 10^{-11}\times 2\times 10^{30}\times 1.9\times 10^{27}}{(7.8\times 10^{11}{)}^2}=-4.17\times 10^{23}\text{ N}$

2. The force exerted by the Sun on an object at a certain distance is 10⁶N. The object travels half the distance to the Sun. What is the force exerted by the Sun on the object now?

$\frac{1}{{\left(\frac{1}{2}\right)}^2}=4$

So, the new force is 4 MN.

3. How much gravitational force do two 1 kg weights 5 cm apart exert on each other?

$F_{grav}=\frac{-GMm}{r^2}=\frac{-6.67\times 10^{-11}\times 1\times 1}{0.05^2}=-2.67\times 10^{-8}\text{ N}$

In other words, ordinary objects exert negligible gravitational force.

4. The radius of the Earth is 6360 km, and its mass is 5.97 x 10²⁴kg. What is the difference between the gravitational force on 1 kg at the top of your body, and on 1 kg at your head, and 1 kg at your feet? (Assume that you are 2m tall.)

$\mathrm{\Delta }{F}_{grav}=GMm(\frac{1}{6360000^2}-\frac{1}{6360002^2})=(1.55\times 10^{-20})GMm=1.55\times 10^{-20}\times 6.67\times 10^{-11}\times 5.97\times 10^{24}\times 1=6.19\mu \text{N}$

This is why it is acceptable to approximate the acceleration due to gravity as constant over small distances.