Torque and Angular Momentum
Studio Physics I
Basics of Torques
Just as was the case for force, torque is a vector and so one must take its direction into
account when applying Newton’s Second Law. Torques which act in opposite directions
are assigned opposite signs (+ or –) and work against one another. Torques which act in
the same direction are assigned the same sign (+ or –) and reinforce one another. Hence,
we need to get good at determining the direction of a torque.


Part I – Determining the Direction of Torque Using   I 
To answer questions regarding direction, DO NOT USE + OR – unless you have clearly
defined what direction you mean when you say + or –. Instead, try to use words like
clockwise, counter clockwise, up, down, right, left, into the page or out of the page.
1. What is the direction of the angular velocity of the disk
shown at the left? (The axis points out of the page.)
2. What is the direction of the angular acceleration of this
disk if it is speeding up?
3. What would the direction be of the torque necessary to
make this disk spin in the direction shown with an
increasing angular speed?
4. What is the direction of the angular acceleration of this
disk if it is slowing down?
5. What would the direction be of the torque necessary to
make this disk spin in the direction shown with an
decreasing angular speed?
6. Consider the pulley shown at the left. (The axis is out of
the page.) There is a mass attached to a rope which is
wrapped around the pulley. If the mass is let free to fall,
what is the direction of the angular velocity of the pulley?
What is the direction of the angular acceleration? What
is the direction of the torque that the mass and rope exert
on the pulley at the point of attachment?
  
  r F
Part II – Determining the Direction of Torque Using
  
7. Use the right hand rule to state the direction of c  a  b for each case shown below.
(Hint: the answer to A is out of the page)
A)
B)
C)
D)
a
b
a
a
b
 1999-2001 K. Cummings; Rev. 2004 Bedrosian
b
a
b
  
8. Use the right hand rule to state the direction of d  b  a for each case shown above.
  
  
What is the relationship between c  a  b and d  b  a ?
9. Consider the pulley shown in question #6. Draw a set of two arrows, one of which
represents the direction of the tension in the string (where it attaches to the pulley) and
  
one which represents the radius of the pulley. Use these two arrows and   r  F to
determine the direction of the torque produced by the tension. Note: The direction of the
r vector is from the axis to the point where the force is applied. However, it will help you
to determine the direction of the cross product if you draw the r and F vectors with their
respective tails at the same point. You can always move a vector when you draw it as
long as you preserve its direction. Compare your answer here to your answer for
question #6 – they should be the same.
Part III – Angular Momentum of a Point Object
10. Suppose you have a stone of mass 200 grams tied to the end of a rope of length
R=0.5 m. The stone is twirled in a horizontal circle of radius 0.5 m as shown in the figure
below. The linear velocity of the stone at one instant is also shown. (But the object
continues to move in circle the next instant.) What is the magnitude and direction of the
  
stone’s angular momentum at this instant? Hint: l  r  p for a point mass.
v
11. Suppose everything about the problem statement above stays the same, except the
rope breaks at the instant shown. Hence, the stone moves in a line rather than a circle in
the moments that follow the instant shown. What is the magnitude and direction of the
stone’s angular momentum at this instant?
12. Consider the stone tied to the end of the rope shown in question #10 above. The rope
is not broken and the stone is moving in a circle. The stone has a angular momentum
(right?). Does the stone have a linear momentum even though it is moving in a circle? If
so, is it constant or changing with time? Explain (or justify) your answer.
13. Now suppose that the rope has broken and the stone is moving is straight line.
(Ignore gravity.) The stone has a linear momentum (right?). Does the stone also have an
angular momentum? Explain (or justify) your answer.
14. When we want to talk about the linear position of an object, we have to choose an
origin with respect to which we measure the position. (That is, we cannot just say “three
meters away”, we have to say “three meters away” from where.) What do you have to
choose to define an angular momentum for an object moving linearly? Why don’t you
have to make this choice when the object moves in circle?
 1999-2001 K. Cummings; Rev. 2004 Bedrosian