Two identical uniform spheres, labeled 1 and 2, each with mass m=9×1015kgm=9×101

Question

Two identical uniform spheres, labeled 1 and 2, each with mass m=9×1015kgm=9×1015kg lie on the y axis equidistant to the origin O as shown above. The distance of a mass to the the origin is d=12kmd=12km. A third identical sphere, labeled by 3 and with the same mass as the first two, is initially placed on the x axis, but very far away (the broken lines mean 'very far away'). Assume that the system is in deep space so that the only forces are gravitational forces between the masses. Assume there is no friction so that energy is conserved. Also assume that spheres 1 and 2 cannot move. Take G=6.67×1011Nm2/kg2G=6.67×10-11Nm2/kg2. Calculate the speed of the third mass when it passes through the origin.

x / (far away) away) 3 2 Two identical uniform spheres, labeled 1 and 2, each with mass m 9 x 1015kg lie on the y axis equidistant to the origin O as shown above. The distance of a mass to the the origin is d 12km. A third identical sphere, labeled by 3 and with the same mass as the first two, is initially placed on the x axis, but very far away (the broken lines mean 'very far away). Assume that the system is in deep space so that the only forces are gravitational forces between the masses. Assume there is no friction so that energy is conserved. Also assume that spheres 1 and 2 cannot move. Take G = 6.67 × 10-11 Nm2/kg2. Calculate the speed of the third mass when it passes through the origin. in 7678558

Explanation / Answer

Given,

m = 9 x 10^15 ; d = 12 km = 12 x 10^3 m ;

We know that

PE = G m1 m2/r^2

We know from conservation of energy

Ei = Ef

U12 = U12 + U23 + U13 + KE

1/2 m v^2 = Gm^2/d + Gm^2/d

v^2 = 4 G m/d

v = sqrt (4 G m/d)

v = sqrt (4 x 6.67 x 10^-11 x 9 x 10^15/(12 x 10^3)) = 14.15 m/s

Hence, v = 14.15 m/s

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