Question

The average distance from the Moon to the center of the Earth is 384,400 km, and the diameter of the Earth is 12,756 km. Calculate the gravitational force that the Moon exerts (a) on a 1-kg rock at the point on the Earth’s surface closest to the Moon and (b) on a 1-kg rock at a point on the Earth’s surface furthest from the Moon. The mass of the Moon is 7.349 × 1022 kg. (c) Find the difference between these two forces. This difference is the apparent tidal force from the Moon pulling the two rocks apart.

Answer 1

a)The gravitational force exerted by the Moon on a 1 kg rock at the point on the Earth closest to the Moon=$F_{1}=GMm/R_{1}^{2}=6.67*10^{-11}Nm^{2}/kg^{2}*7.349*10^{22}kg*1kg/(D-d/2)^{2}m^{2}=6.67*7.349*10^{11}/((384400-12756/2)*10^{3})^{2}=3.5*10^{-5} N.$

where G is the gravitational constant, M is the mass of the Moon, m is 1 kg, and R₁ is the distance from the Moon to the 1 kg rock=Distance from the Moon to the center of the Earth D-radius of the earth i.e d/2 in meters.

b)Similarly the gravitational force exerted by the Moon on a 1 kg rock at the farthest point on the Earth=$F_{2}=GMm/R_{2}^{2}=6.67*10^{-11}Nm^{2}/kg^{2}*7.349*10^{22} kg*1kg/(D+d/2)^{2}m^{2}=6.67*7.349*10^{11}/(384400+12756/2)^{2}*10^{6}=3.2*10^{-5} N.$

where R₂=D+d/2 i.e the distance from the Moon to the 1 kg rock in meters.

c)The difference between these 2 forces $F=F_{1}-F_{2}=(3.5-3.2)*10^{-5} N=0.3*10^{-5} N.$