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PHY221 Ch15: // Axis Theorem and Torque

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. PHY221 Ch15: // Axis Theorem and Torque. Main Points.

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PHY221 Ch15: // Axis Theorem and Torque

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  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: CM  vCM 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/cmby 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 Theoremvia K=Kcm+1/2Iw2 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, . 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: F  m y r x O

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

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