Question

Earth is 149.6 billion meters from the sun and takes 365 days to make one complete revolution around the sun. mars is 227.9 billion meters from the sun and has an orbital period of 687 days. what is the ratio of earth's centripetal acceleration to mars's centripetal acceleration?

Answer 1

2.33

Centripetal acceleration is calculated with ac=v2r where v is the orbital speed and r is the radius of the orbit. First, find the orbital speed using the distance traveled in one period: v=2πrT where 2πr is the circumference of the orbit and T is the period. The centripetal acceleration can then be expressed as ac=4π2rT2. The ratio of Earth’s acceleration to Mars’s acceleration is ac,Eac,M=rE/T2ErM/T2M=149.6×109 m/(365 d)2227.9×109 m/(687 d)2=2.33.

Answer 2

We need to assume that the orbits are circular; this is a very good approximation and do not hesitate to adopt it. We need then to consider the basic equation of circular motion that relates centripetal acceleration, angular velocity, mass, radius and force:
$F=m*v^2*r$ . We also have the basic equation of Newton, F=ma. By checking, we see that for each planet we have that a=v^2*r where v is the angular velocity. v=$\frac{2\pi}{T}$ where T is the orbital period, hence we have that a=$\frac{4\pi^2*r}{T^2}$. Denote with the index 1 the Earth case and with 2 the Mars case. We have then that $\frac{a_1}{a_2} = \frac{4 \pi^2*r_{1}*T_2^2}{4 \pi^2*r_{2}*T_1^ 2}$. Substituting the known values in a calculator, we get the result: the centripetal acceleration of earth is 2.32 times greater than that of Mars.