Some examples of torque as a vector

Let's do a few simple examples.

1. Barb the Plumber is hard at work removing a clogged pipe. As she places her wrench around the head of the bolt, the 30-cm handle sticks vertically upwards. Barb then pulls on the end of the handle with a force of 250 N to the South.

   What is the torque exerted by Barb around the head of the bolt?

2. The engine of a spacecraft sticks out from the center of the ship on a long spar. Its position relative to the center of the ship is

   
             r  =  ( 32  m) i  +   ( 22  m) j   -  ( 13  m) k
       

   A passing asteroid exerts a gravitational force on the engine:

   
             F  =  ( 0.2 N) i  -   ( 0.1 N) j   +  ( 1.2  N) k
       

   What is the torque on the engine around the center of the spaceship?

3. 

   A ball of mass M = 3 kg is attached to a massless rod of length L = 6 m. The rod is attached to a pivot and is initially horizontal. Fred holds everything in place.

   1. What is the moment of inertia of the arm-plus-ball around the pivot?

   Fred now releases the ball and rod, which start to swing downwards. A moment later, the rod has swung by an angle β from the horizontal.

   
   1. What is the torque on the ball due to gravity at this moment?
   2. If the ball swings just a teeny little extra angle dβ, how much work will the torque do during that teeny motion?
   3. Write down -- but don't evaluate -- an integral which expresses the amount of work done by the torque as the rod swings from its starting point to an angle of 45 degrees.
   4. Write down -- and DO evaluate -- an integral showing the work done by the torque as the rod rotates from its starting point to its lowest point.
   5. What is the angular velocity of the rod when it reaches its lowest point?
   6. What is the linear velocity of the ball when it reaches its lowest point?
   7. Is there any easier way to do this particular problem?