A spherical planet of radius has a gravitational field strength of magnitude at its…

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Newtonian gravitation as a field theory · Special relativity: spacetime and kinematics

Relativity and Gravitation, Ch Tidal forces and the Newtonian tidal field

Problem

A spherical planet of radius

R = 3.00\times10^{6}\ \mathrm{m}

has a gravitational field strength of magnitude

6.00\ \mathrm{m\,s^{-2}}

at its surface. A sealed capsule falls freely, engines off, at the moment when its centre is at

r = 6.00\times10^{6}\ \mathrm{m}

from the planet's centre. Inside it, two pairs of small beads are released at rest relative to the capsule: pair A separated by

2.00

m along the radial direction, pair B separated by

2.00

m at right angles to the radius. The capsule clock reads a lapse of

120

s between release and measurement, and over that lapse the relative accelerations may be treated as constant and the capsule's local frame as inertial. Air inside the capsule is absent and the beads do not touch anything. Take

c = 2.998\times10^{8}\ \mathrm{m\,s^{-1}}

. ((a)) Find the mass

M

of the planet. ((b)) Find the change in the separation of pair A after the

120

s of capsule time, and say whether it is an increase or a decrease. ((c)) Find the change in the separation of pair B, and say whether it is an increase or a decrease. ((d)) A second capsule passes the first at a constant speed

0.600c

relative to it. What lapse of time does the second capsule's frame assign to the same experiment, from release to measurement?

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