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22 February, 13:19

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.

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  1. 22 February, 15:00
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    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
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