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Topic : Moment of Inertia

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1 Topic : Moment of Inertia
PARUL INSTITUTE OF ENGINEERING & TECHNOLOGY CIVIL DEPARTMENT Topic : Moment of Inertia

2 Name of Students Hathila Jay : 130370106052
Parmar Shitul : Pargi Bharat : 4. Parmar Ronak : 5.Mal pankaj: 6.Vasaiya Priyakant:

3 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.

4 Two Spheres 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 m m r

5 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 Dm. The total moment of inertia is Each mass element contributes The sum becomes an integral distance r to r+Dr length L axis

6 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

7 Playground Ride 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 m M r

8 Parallel Axis Theorem Some objects don’t rotate about the axis at the center of mass. The moment of inertia for a rod about its center of mass: Some objects don’t rotate about the axis at the center of mass. The moment of inertia depends on the distance between axes. 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 Spinning Coin 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 R M M R Id Ie next

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