Lecture 21: Torques - Oscillating Bodies - Hoops
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1. Key Equations for Angular Momentum and Torque:
A review is given of equations for angular momentum and torque, and the importance of choosing the point of origin. These equations are exercised using an example of a circular orbit. If the point of origin is placed at the center of the circle, angular momentum is conserved; for any other point, angular momentum is not conserved.
2. Dynamics of a Spinning Rod:
Two examples are analytically described: (1) a rod spinning around an off-center pin and (2) a rod spinning around its center of mass. A third example is discussed in great detail: we hit a rod, initially at rest, on a frictionless horizontal table. The resulting motion can be viewed as a linear motion (with constant speed) of the center of mass and a rotational motion around the center of mass. The discussion concludes with a demo of a ruler being struck (at various places) on a horizontal surface (though not frictionless).
3. Physical Pendulum:
A ruler is suspended from a pin offset from the ruler's center of mass to form a pendulum. The equation of motion is derived by analyzing the torque and moment of inertia using the pin as the point of origin. For small angles, the motion is simple harmonic. This analysis is also conducted for a hoop suspended from a pin. When gravity is the only restoring force, the period of such a pendulum is determined solely by geometry and is independent of mass. The predicted periods of these pendulums are quantitatively confirmed in demos.
4. Friction, Kinetic Energy and a Spinning Top:
Remember the bizarre top that kept spinning during lecture 18? Maybe there was something hidden in that black box. A completely new challenge is now to explain the bizarre behavior of a spinning blue plastic object which seems to defy the laws of physics. Another brain teaser!
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