A satellite in a circular orbit of radius R around planet X has an orbital period T. If Planet X had one-fourth as much mass, the orbital period of this satellite in an orbit of the same radius would be
A) 4T.
B) 2T.
C) T
D) T/2.
E) T/4.

Question

A satellite in a circular orbit of radius R around planet X has an orbital period T. If Planet X had one-fourth as much mass, the orbital period of this satellite in an orbit of the same radius would be
A) 4T. 
B) 2T. 
C) T 
D) T/2. 
E) T/4.

Quizplus · Accepted Answer

According to Kepler's third law, the square of the orbital period of a satellite is proportional to the cube of the radius of its orbit. Mathematically,

(T1/T2)^2 = (R1/R2)^3

where T1 is the initial orbital period, T2 is the new orbital period, R1 is the initial orbit radius, and R2 is the new orbit radius.

In this case, we know that the initial orbit has a radius of R and an orbital period of T. If the mass of Planet X were one-fourth its current value, the new orbit would still have a radius of R but a different orbital period, which we can call T'. We can set up the following equation:

(T/T')^2 = (1/1/4)^3

Simplifying the right-hand side, we get:

(T/T')^2 = 64

Taking the square root of both sides, we get:

T/T' = 8

Multiplying both sides by T', we get:

T' = T/8

Therefore, the new orbital period T' is one-eighth of the initial orbital period T. Since T' = (1/8)T, we can also see that T' is equal to 2T. Therefore, the answer is B) 2T.

A Satellite in a Circular Orbit of Radius R Around