12.62 The magnitude of the tidal force exerted on an object of mass m and length

Question

12.62

The magnitude of the tidal force exerted on an object of mass m and length a is approximately 4GmMa/r3.
In this expression, M is the mass of the body causing the tidal force and r is the distance from the center of m to the center of M. Suppose you are 1.0×106 mi away from a black hole whose mass is a 1.2×106 times that of the Sun. Estimate your mass to be 71 kg and your length to be 1.7 m .

Part A

Estimate the tidal force exerted on your body by the black hole.

Express your answer using two significant figures.

F_tidal=?

Part B

At what distance will the tidal force be approximately 14 times greater than your weight?

Express your answer using two significant figures.

r=?

Please explain answer ! Thank you

Explanation / Answer

Given that F = 4GmMa/r^3

G is the univeral gravitational constant

m = 71 kg

M= 1.2*10^6*M_sun = 1.2*10^6*1.98*10^30 = 2.376*10^36 kg

a = 1.7 m

r = 1*10^6 mi = 1*10^6*1609.34 = 1.609*10^9 m

then tidal force is F = 4GmMa/r^3

F =(6.67*10^-11*71*2.376*10^36*1.7)/(1.609*10^9)^3

F = 4.59 N

---------------------------------------------------------

B) now F2 = 14*71*9.81 = 9751.14 N

F1/F2 = (r2/r1)^3

F1 = 4.59 N

F2 = 9751.14 N

r1 = 1.609*10^9 m

r2 = ?

4.59/9751.14 = (r2/1.609*10^9)^3

r2/(1.609*10^9) = 0.0777

r2 = 1.25*10^8 m

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