1 / 10

Moment of Inertia

Moment of Inertia. 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 = 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.

jane-ashley
Download Presentation

Moment of Inertia

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related