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. A cube of mass M and side length L is placed with one vertex at the origin, as shown in the diagram below. That vertex, at the origin, attached to a universal joint which can rotate in all directions.

   

   
   
      Q:  What is the location of the center of mass of the cube? 
   
      Q:  What is the torque due to gravity on the cube around the origin? 
   
   
   
   
   
   
   
   

3. 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?

4. Joe the Artist designs a minimalist chandelier. It consists of a small black sphere which sticks into the ceiling, and four red lights that hang down from the sphere. Joe labels the lights "a", "b", "c", "d". Each one has mass 6 kg. The lights are attached to a set of massless rods of length 2 m which form a square. Green wires connect the lights to the black sphere at the top of the unit, which is 3 m above the square.

   

   Use a coordinate system in which the black sphere sits at the origin, as shown in the figure. Note that the lights hang down in the negative-Z direction.

   1. What is the gravitational force on light "a"? Express as a vector.
   2. What is the position of light "a"? Express as a vector.
   3. What is the torque around the sphere due to the gravitational force on light "a"? Express as a vector.
   4. What is the net gravitational force on all the lights?
   5. What is the net gravitational torque around the black sphere due to all the lights?

5. 

   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 = 10 degrees from the horizontal.

   
   2. What is the torque on the ball due to gravity at this moment?
   3. What is the angular acceleration of the arm-plus-ball at this moment?

   Eventually, the arms swings so that it is pointing straight down.

   
   4. What is the torque on the ball due to gravity at this moment?
   5. What is the angular acceleration of the arm-plus-ball at this moment?

   As the arm continues to swing, it starts to rise up on the left-hand side of the pivot.

   
   6. What is the torque on the ball due to gravity at this moment?
   7. What is the angular acceleration of the arm-plus-ball at this moment?