Question

All the planets we know of rotate, some fast and some slow. Is there any limit on how fast they can rotate, or could it in principle be any speed? In this problem we will answer that question! Suppose (to keep it simple) that a particular planet is spherical and has a uniform mass density ρ. Consider a rock resting on the surface of that planet, at the equator. (a) Draw a free-body diagram of that rock, and write down Newton’s second law for the rock, putting in all your forces. (You can neglect the motion of the planet around its sun, but do NOT neglect the fact that it is travelling in a circle as the planet rotates about its own axis!) (b) Find an algebraic expression for the minimum value that the period of rotation T could have, physically, using your result from part (a). Comment intelligently on your solution. (c) Evaluate your expression for T if ρ = 5.5 g/cm³ (the average density of the Earth.) Say something intelligent about the numbers you got. (d) Estimate the size of the error we made in the case of the Earth by neglecting the Earth’s motion around the Sun. Was it a good approximation? Say something intelligent.

Answer 1

a) Free body diagram of rock at the equator:

b) Net force on the rock is ( m is mass of rock, g is acceleration due to gravity, R is radius of earth , is angular velocity of earth,N is normal reaction on rock by earth)

Minimum value of period of rotation is

When the planet has this rotation time period ,the rocks on the surface of the planets at the equator will be just floating in air above the surface of the planet.

c)

Earth's average density is , For the rock on earth's equator to float just above the earth surface , earth needs to complete a day in 1.4 hours.

d)

In case of earth, time period of rotation is

Accepted value of acceleration due to gravity

Due to rotation, the centrifugal acceleration is

Since we are neglecting the rotation of earth, error in accepted acceleration due to gravity is

Since , so it is a good approximation .