Imagine that Earth were in a stable orbit at its current distance from the Sun, but around a different star that had twice the mass of the Sun. Which of the following would have to be true? A. Earth's year would be shorter.
B. Earth's year would be longer.
C. Earth's mass would be lower.
D. Earth's mass would be higher.

Explanation:

We want to know if the mass of the earth and the sun affects the rotation of the earth.

Mass is a constant, so the earth mass cannot change, then option C and D are nullified

Now, for the period of oscillation.

Now the force of attraction is given as

F=kM1M2/r²

Me= Mass of earth

Ms = mass of sun

Mn= mass of star

Mass of star is twice mass of sun

Mn=2Ms.

Force between the earth and sun

F1=kMeMs/r²

Force between the earth and star

F2=kMeMn/r²

k, r and Me did not change

F1/Ms=F2/Mn

Since Mn=2Ms

Then, F1/Ms=F2/2Ms

F1=F2/2

Then, F2=2F1

The force attraction between The earth and the star is twice the earth and the sun

The force keeping the earth in circular motion is

F=mv²/r

Therefore,

Since the earth has a constant mass and same radius from both of them

Then F/v²=k

F1/v1²=F2/v2²

F2=2F1

F1/v1²=2F1/v2²

Therefore, v2²=2v1²

v2=v1√2

The velocity of the earth to star is higher than velocity of earth to sun

Comparing it to frequency it is given that

w=vr

2πf=vr

Radius and 2π are constant, them we have

F1/V1=F2/V2

V2=v1√2

F1/V1=F2/v1√2

F1=F2/√2

Then, f2=f1√2

The frequency of the earth to star is higher than earth to sun.

This implies that it will be faster to complete the cycle than the earth to sun

Therefore the earth year will be shorter is