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Rotational Inertia

Rotational Inertia. Circular Motion. Objects in circular motion have kinetic energy. K = ½ m v 2 The velocity can be converted to angular quantities. K = ½ m ( r w ) 2 K = ½ ( m r 2 ) w 2. m. r. w. Integrated Mass .

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Rotational Inertia

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  1. Rotational Inertia

  2. Circular Motion • Objects in circular motion have kinetic energy. • K = ½ m v2 • The velocity can be converted to angular quantities. • K = ½ m (rw)2 • K = ½ (mr2) w2 m r w

  3. Integrated Mass • The kinetic energy is due to the kinetic energy of the individual pieces. • The form is similar to linear kinetic energy. • KCM = ½ mv2 • Krot = ½ Iw2 • The term Iis the moment of inertia of a particle.

  4. Moment of Inertia Defined • The moment of inertia measures the resistance to a change in rotation. • Mass measures resistance to change in velocity • Moment of inertia I = mr2 for a single mass • The total moment of inertia is due to the sum of masses at a distance from the axis of rotation.

  5. A spun baton has a moment of inertia due to each separate mass. I = mr2 + mr2 = 2mr2 If it spins around one end, only the far mass counts. I = m(2r)2 = 4mr2 Two Spheres m m r

  6. Extended objects can be treated as a sum of small masses. A straight rod (M) is a set of identical masses Dm. The total moment of inertia is Each mass element contributes The sum becomes an integral Mass at a Radius distance r to r+Dr length L axis

  7. Rigid Body Rotation • The moments of inertia for many shapes can found by integration. • Ring or hollow cylinder: I= MR2 • Solid cylinder: I= (1/2)MR2 • Hollow sphere: I= (2/3)MR2 • Solid sphere: I= (2/5)MR2

  8. The point mass, ring and hollow cylinder all have the same moment of inertia. I= MR2 All the mass is equally far away from the axis. The rod and rectangular plate also have the same moment of inertia. I= (1/3) MR2 The distribution of mass from the axis is the same. Point and Ring M R R M M M length R length R axis

  9. Some objects don’t rotate about the axis at the center of mass. The moment of inertia depends on the distance between axes. The moment of inertia for a rod about its center of mass: Parallel Axis Theorem h = R/2 M axis

  10. How much energy is stored in the spinning earth? The earth spins about its axis. The moment of inertia for a sphere: I = 2/5 MR2 The kinetic energy for the earth: Krot = 1/5 MR2w2 With values: K = 2.56 x 1029 J Spinning Energy The energy is equivalent to about 10,000 times the solar energy received in one year. next

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