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Moment of Inertia

Moment of Inertia. Moment of Inertia. Units: kg m 2 The quantity ∑m i r i 2 , which is the proportionality constant between angular acceleration and net torque Torque Equation Depends on the axis of rotation. Torque— τ net Moment of Inertia—I Angular Acceleration— α

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Moment of Inertia

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  1. Moment of Inertia

  2. Moment of Inertia • Units: kg m2 • The quantity ∑miri2, which is the proportionality constant between angular acceleration and net torque • Torque Equation • Depends on the axis of rotation

  3. Torque—τnet Moment of Inertia—I Angular Acceleration—α Newton’s Second Law—τnet=I α Force—Fnet Mass—m Acceleration—a Newton’s Second Law—Fnet=ma Angular Versus Linear

  4. Moment of Inertia • Newton’s second law is easy to write, but we need to know the object’s moment of inertia. • Unlike mass, we cannot measure the moment of inertia. We must calculate it. • The moment of inertia is the sum over all the particles of the system.

  5. Moment of Inertia Where r is the distance from the rotation axis. If we let the axis of rotation be the z-axis then the moment of inertia becomes:

  6. Problem 1 • Find the moment of inertia of a thin, uniform rod of length L and mass M that rotates about a pivot at one end.

  7. Answer Problem 1

  8. Problem 2 • Find the moment of inertia of a circular disk of radius R and mass M that rotates on an axis passing through its center.

  9. Answer Problem 2

  10. Parallel-Axis Theorem • The moment of inertia depends on the axis of rotation. • If you need to know the moment of inertia for an axis that is not through the center of mass you can use an axis parallel to that and the parallel-axis theorem. Where d is the distance from the axis to the axis through the center of mass.

  11. Problem 3 • The engine in a small airplane is specified to have a torque of 60Nm. This engine drives a 2.0 m long, 40 kg propeller. On start-up, how long does it take the propeller to reach 200 rpm?

  12. Answer Problem 3 • The propeller can be considered a rod that rotates about its center. • The moment of inertia of a rod about its center is 1/12 ML2. • I = 1/12(40 kg)(2.0 m)2 = 13.33 kg m2 • α = τ/I = 60Nm/13.33kgm2 = 4.50 rad/s2

  13. Problem 4 • A 10.0 cm diameter, 5.0 kg disk turns on an axle. A vertical cable attached to the edge of the disk exerts a 100 N force but, initially, a pin keeps the disk from rotating. What is the initial angular acceleration of the disk when the pin is removed? 100N Axle mg

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