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This contains Quality notes on Moment of Inertia. …

This contains Quality notes on Moment of Inertia.
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