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THANKS TO SINCERE EFFORTS OF MR. SURYAVANSHI PARTH & MR. HARDIK PATEL.

& PROF. SOHEL PATEL

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- 1. CHAPTER 9 MOMENT OF INERTIA Faculty Name:HARDIK PATEL Dept. MECH LJ Polytechnic
- 2. Moment of inertia Definition 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 Faculty Name Dept. LJ Polytechnic N i iirmI 1 2
- 3. Moment of inertia How to calculate M.I 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 Faculty Name Dept. LJ Polytechnic m r m
- 4. Moment of inertia M.I of straight bar Faculty Name Dept. LJ Polytechnic The total moment of inertia is Each mass element contributes The sum becomes an integral 2 )( rmI rrLMI rLMm 2 )/( )/( 23 0 2 )3/1()3/)(/( )/( MLLLMI drrLMI L Extended objects can be treated as a sum of small masses. A straight rod (M) is a set of identical masses Dm. axis length L distance r to r+r
- 5. Moment of inertia M.I for different shapes Faculty Name Dept. LJ Polytechnic 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. Moment of inertia Parallel axis theorem Faculty Name Dept. LJ Polytechnic Some objects don’t rotate about the axis at the center of mass. The moment of inertia depends on the distance between axes. 2 MhII CM The moment of inertia for a rod about its center of mass 2 22 22 )12/1( )4/1()3/1( )2/()3/1( MRI MRMRI RMIMR CM CM CM
- 7. Moment of inertia Perpendicular axis theorem Faculty Name Dept. LJ Polytechnic 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. yxz III M Ix = (1/12) Mb2 Iy = (1/12) Ma2 a b Iz = (1/12) M(a2 +b2)
- 8. Moment of inertia Radius of gyration Faculty Name Dept. LJ Polytechnic • Consider area A with moment of inertia Ix. Imagine that the area is concentrated in a thin strip parallel to the x axis with equivalent Ix. A I kAkI x xxx 2 kx =radius of gyration with respect to the x axis Similarly, A J kAkJ A I kAkI O OOO y yyy 2 2 222 yxO kkk
- 9. Moment of inertia M.I for standard shapes Faculty Name Dept. LJ Polytechnic
- 10. Moment of inertia M.I for standard shapes Faculty Name Dept. LJ Polytechnic

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