Question

1. The Earth spins one full rotation every 24 hours. You, a mad scientist bent on destroying the world, wish to make "Earth stand still" - you want to make it stop rotating. In your arsenal, you have lots and lots of rocket engines that you are free to place however you wish around the Earth. How will you arrange these rockets to stop the Earth's rotation (where will you put them, how will you orient them)? How much total thrust will you need to stop the rotation? Assume the Earth is a uniform sphere. You can find the mass and radius of the Earth, and you will need the moment of inertia through a sphere's center of mass, which you can also find on Google.

2. When an object rolls without slipping, the static friction acts antiparallel to the direction of motion. Why don't we usually worry about the (negative) work done by the static friction on the rolling body? Hint: How does the rotational work done by the static friction compare? It will help to set up an example problem and calculate the quantities explicitly to compare them.

Answer 1

mass of earth M = 5.97e+24 kg

radius R = 6378 km

we take earth as sphere

MI about its axis = 2/5 MR² = 9.71e+37 kg-m²  

period of earth rotation T = 24 hrs

angular speed   = 2/T = 7.27e-5 rads/s

earth rotates from WEST to EAST, the rockets have to be fired in the EAST direction it will deliver some momentum to earth in the WEST direction and oppose the motion of earth.

Suppose F is the force the rocket exerts on the surface of earth

torque = FR ; R is radius of earth

= I ;   is angular acceleration given to earth in opposite direction of its motion

is to be comparable to   so that will be 0 in a comparable time of few days if not in seconds or hours.

we take = 7.0e-10 , earth stops in about a day

Thrust required F = I/R = 9.7e+37 * 7.0e-10/6.4e+6

= 1.0 e+ 22 N

2. When the body is rolling , the point of contact is momentarily stationary. Frictional force act at the point of contact. The force does work when the point of action is displaced. In this case the frictional force does not do any work as the point of action is not displaced.

In the case of rotation the force causes a torque on the COM and makes the body rotates about the point of contact.