1. Moments of Inertia and Rolling cylinders Prof. Miles Padgett FRSE Dept. Physics and Astronomy University of Glasgow m.padgett@physics.gla.ac.uk

2. Moment of Inertia Moment of inertia of an object ( I ) Moment of inertia of an object ( I ) –rotational energy = 1 / 2 I  2 –angular momentum = I  Moment of inertia =  m i r i 2 Moment of inertia =  m i r i 2 m1m1 r1r1 r2r2 m2m2 r3r3 m3m3

3. Moment of inertia of a solid cylinder n Moment of inertia of a thin ring I = Mr 2 =  r  r  l r 2 rr r n Integrate to get a solid cylinder r I = ∫ 0 R  l r 3  r = 1 / 4  l R 4 = 1 / 2 mR 2 = 1 / 2 mR 2 R l

4. Moment of inertia of a hollow cylinder n Moment of Inertia of a solid cylinder n A hollow cylinder I = 1 / 2 mR 2 = - m1m1 R1R1 R2R2 m2m2 I = 1 / 2 m 1 R 1 2 - 1 / 2 m 2 R 2 2 = 1 / 2 M (R 1 2 + R 2 2 ) M

5. How fast does a cylinder roll? n Equate energy at top of hill to energy at bottom Potential energy Kinetic energy + rotational energy h M v  E = Mgh = 1 / 2 Mv 2 + 1 / 2 I  2 potentialkineticrotational

6. But this gets “simpler” Mgh = 1 / 2 Mv 2 + 1 / 2 I  2 Substitute for I = 1 / 2 M (R 1 2 + R 2 2 ) Substitute for I = 1 / 2 M (R 1 2 + R 2 2 ) Mgh = 1 / 2 Mv 2 + 1 / 4 M (R 1 2 + R 2 2 )  2 Substitute for  = v/r 1 Substitute for  = v/r 1 Mgh = 1 / 2 Mv 2 + 1 / 4 M (R 1 2 + R 2 2 )(v/R 1 ) 2 potential kinetic rotational

7. ….or (if not simpler) at least more useful! n Note, mass cancels from each term gh = 1 / 2 v 2 + 1 / 4 (R 1 2 + R 2 2 )(v/R 1 ) 2 n Re-arrange for v v = 2gh n Note velocity depends NOT on mass, only on SHAPE As hole (R 2 ) gets bigger hollow cylinder goes slower As hole (R 2 ) gets bigger hollow cylinder goes slower 1+ 1 / 2 (1 + (R 2 /R 1 ) 2 )

8. Which cylinder rolls faster? n The solid cylinder rolls faster

9. Which material rolls faster? n Both materials roll at the same speed