Equilibrium,Torque and
Angular Momentum
Lecture 10
Tuesday: 17 February 2004
Defining Rotational Inertia
•The larger the mass, the smaller the
acceleration produced by a given force.


F  ma
•The rotational inertia I plays the equivalent role
in rotational motion as mass m in translational
motion.
•I is a measure of how hard it is to get an object
rotating. The larger I, the smaller the angular
acceleration produced by a given force.
Determining the Rotational
Inertia of an Object
I is a function of both the mass and shape of the object.
It also depends on the axis of rotation.
1. For common shapes, rotational inertias are listed in tables. A
simple version of which is in chapter 11 of your text book.
2. For collections of point masses, we can use :
i N
I   mi ri 2
i 1
where r is the distance from the axis (or point) of rotation.
3. For more complicated objects made up of objects from #1 or
#2 above, we can use the fact that rotational inertia is a scalar
and so just adds as mass would.
Torque as a Cross Product
 
  r F
  
  r F sin 

(Like F=Ma)
The direction of the Torque is always in the direction of
the angular acceleration.
• For objects in equilibrium, =0 AND F=0
Torque Corresponds to Force
• Just as Force produces translational acceleration
(causes linear motion in an object starting at rest,
for example)
• Torque produces rotational acceleration (cause a
rotational motion in an object starting from rest,
for example)
• The “cross” or “vector” product is another way to multiply
vectors. Cross product results in a vector (e.g. Torque).
Dot product (goes with cos ) results in a scalar (e.g. Work)
• r is the vector that starts at the point (or axis) of rotation and
ends on the point at which the force is applied.
Does an object have to be moving in a
circle to have angular momentum?
• No.
• Once we define a point (or axis) of rotation (that
 is,
a center), any object with a linear momentum p
that does not move directly through that point has
an angular momentum defined relative to the
chosen center as   
Lrp