1. PHY221 Ch15: // Axis Theorem and Torque
Recall main points:
Expression of the Kinetic energy of a rigid body in terms of Kcm and Icm
Parallel Axis Theorem
Torque and Cross Product
Angular acceleration from the torque

2. PHY221 Ch15: // Axis Theorem and Torque
Main Points
Kinetic energy of a rigid body in terms of Kcm and Icm
In the 3rd slide of Ch 13 we showed that the total kinetic energy of a system is equal to the energy of the Center of Mass (CM) plus the energy RELATIVE to the CM:
vCM  CM
In our case since we study RIGID bodies, any motion relative to the CM MUST be a rotation around an axis through the CM. And therefore, since :
This is a general result that should be used when the work-energy theorem, or the conservation of mech. energy, is used for solving problems involving rigid bodies rotating . Note that any motion can be viewed as the CM motion PLUS a rotation AROUND an axis thru the CM

3. PHY221 Ch15: // Axis Theorem and Torque
Main Points
Optional: Direct proof of the preceding result:
By definition of the center of mass (see CM chapter) the last term is zero. Note also that we replaced vi/cm by ri/cm since we are dealing with a rigid body. We obtain the fundamental result:

4. PHY221 Ch15: // Axis Theorem and Torque
Main Points
Parallel Axis Theorem via KO = Kcm+ 1/2 Icm w2
Using the result on the previous slides, we can arrive at the very useful parallel axis theorem:

5. PHY221 Ch15: // Axis Theorem and Torque
Main Points
Torque and Cross product:
The torque is defined as:
Right Hand Rule:
In this course we only look at rotations around a axis of fixed direction, therefore, from studying the right hand rule for a second, we see that the only components of r and F that matter are the x and y components (see picture at right). In the problems you are given, the forces are always in the X-Y plane and the r should always be the vector r=(x,y) and thus  is the angle in the x-y plane. The magnitude of the torque around the z-axis is: x y F  r m O

6. PHY221 Ch15: // Axis Theorem and Torque
Main Points
Angular acceleration from the torque, using Newton’s 2nd law:

7. PHY221 Ch15: // Axis Theorem and Torque
Assignment F r F r
Take positive values for vectors directed into the page
1. rF –rF rF/ NPA CM
w/O=?
w/CM
-w/CM
NPA O q CM
Parallel Axis theorem requires:
Only // axes I/O=I/O’ +Md2 for any O and O’
// Axes and O’=CM
// axes or O’=CM