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Rotational Inertia

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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 ) 2 K = ½ (m r 2 ) 2 r m

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Integrated Mass The kinetic energy is due to the kinetic energy of the individual pieces. The form is similar to linear kinetic energy. K CM = ½ m v 2 K rot = ½ I 2 The term I is the moment of inertia of a particle.

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Moment of Inertia Defined The moment of inertia measures the resistance to a change in rotation. Mass measures resistance to change in velocityMass measures resistance to change in velocity Moment of inertia I = mr 2 for a single massMoment of inertia I = mr 2 for a single mass The total moment of inertia is due to the sum of masses at a distance from the axis of rotation.

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

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

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

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Point and Ring The point mass, ring and hollow cylinder all have the same moment of inertia. I = MR 2I = MR 2 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) MR 2 The distribution of mass from the axis is the same. R M M R axis length R M M

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Parallel Axis Theorem 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: axis M h = R/2

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Spinning Energy 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 M R 2 The kinetic energy for the earth: K rot = 1/5 M R 2 2 With values: K = 2.56 x 10 29 J The energy is equivalent to about 10,000 times the solar energy received in one year. next

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Summary Lecture 11 Rotational Motion 10.5Relation between angular and linear variables 10.6Kinetic Energy of Rotation 10.7Rotational Inertia 10.8Torque.

Summary Lecture 11 Rotational Motion 10.5Relation between angular and linear variables 10.6Kinetic Energy of Rotation 10.7Rotational Inertia 10.8Torque.

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