The next three questions pertain to the following situation. Planet X of mass m1 orbits a Sun in uniform circular motion at a distance r1 and speed v1. The mass of the Sun is MS1 and its radius is RS1 as shown in the figure below.

http://https://smart2.physics.illinois.edu/Content/Media/Images/Phys100/hb08/OrbitPlanetX.gif

If the planet's mass were doubled, how would its speed v1 change?

v1 would remain the same
v1 would increase by a factor of the square root of 2
v1 would increase by a factor of 2


Consider another solar system in which Planet Y of mass m2 = 2 m1 orbits a Sun of mass MS2 = 4 MS1 and radius RS2 = 2RS1 at a distance r2 = 2r1. Further, suppose R2, the radius of Planet Y is twice R1, the radius of Planet X.

w2 < w1
w2 = w1
w2 > w1


How does v2, the speed of Planet Y in its orbit, compare to v1, the speed of planet X in its orbit?

v2 = (1/sqrt(2))v1
v2 = (1/2)v1
v2 = v1
v2 = sqrt(2)v1
v2 = 2v1

All of these can be answered by considering that the force of gravity between sun and planet must equal the centripedal force on the planet:


\frac {GM_1m_2} {R^2} = m_2 \omega ^2 R


which after a bit of rearranging gives:


G m_1 = \omega^2 R^3


Note that the mass of the planet, m_2, has no bearing on the equation of motion. So use this, plus the fact that the velocity of the planet is v_2 = \omega R, to answer the questions.