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## Rotational Inertia

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**Real objects have mass at points other than the center of**mass. Each point in an object can be measured from an origin at the center of mass. If the positions are fixed compared to the center of mass it is a rigid body. Rigid Body ri**The motion of a rigid body includes the motion of its center**of mass. This is translational motion A rigid body can also move while its center of mass is fixed. This is rotational motion. Translation and Rotation vCM **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**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.**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.**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**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**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**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**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**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.