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Moment of Inertia

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  1. Moment of Inertia

  2. Moment of Inertia Defined • The moment of inertia measures the resistance to a change in rotation. • Change in rotation from torque • 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.

  3. 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

  4. 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

  5. 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

  6. 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

  7. A child of 180 N sits at the edge of a merry-go-round with radius 2.0 m and mass 160 kg. What is the moment of inertia, including the child? Assume the merry-go-round is a disk. Id = (1/2)Mr2 = 320 kg m2 Treat the child as a point mass. W = mg, m = W/g = 18 kg. Ic = mr2 = 72 kg m2 The total moment of inertia is the sum. I = Id + Ic = 390 kg m2 Playground Ride m M r

  8. 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

  9. Perpendicular Axis Theorem • For flat objects the rotational moment of inertia of the axes in the plane is related to the moment of inertia perpendicular to the plane. Iy= (1/12) Ma2 b Ix= (1/12) Mb2 M a Iz= (1/12) M(a2 +b2)

  10. What is the moment of inertia of a coin of mass M and radius R spinning on one edge? The moment of inertia of a spinning disk perpendicular to the plane is known. Id= (1/2) MR2 The disk has two equal axes in the plane. The perpendicular axis theorem links these. Id= Ie+ Ie= (1/2) MR2 Ie= (1/4) MR2 Spinning Coin R M M R Id Ie next