Two cylindrical space stations, the second four times the diameter of the first, rotate so as to provide the same amount of artificial gravity. If the first station makes one rotation in the time T, then the second station makes one rotation in time A. T/4 B. 2T C. 4T D. 16T

B. T1 = 2T2

Explanation:

Given

r2 = 4r1

In this case, the centripetal forces must be equal (a1 = a2) for a person on the inside of the outer wall to feel the same amount of artificial gravity.

For a uniform circular motion, v = 2πr/T

Since a1 = a2 and a = v²/r

Then

(v1)²/r1 = (V2)²/r2

(2πr1/T)²/r1 = (2πr2/T)²/r2

4π(r1)²/(r1T1²) = 4π(r2)²/(r2T2²) divide both sides by 4π

(r1)²/r1T1² = (r2)²/r2T2²

r1/T1² = r2/T2² make T2 the subject of formula

T2² = r2 * T1²/r1 Square root both sides

T2 = T1√(r2/r1)

Substitute 4r1 for r2

T2 = T1√(4r1/r1)

T2 = T1√4

T2 = 2T1

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