Moments of Inertia and Rolling cylinders

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Moments of InertiaandRolling cylinders

• Prof. Miles Padgett FRSE
• Dept. Physics and Astronomy
• University of Glasgow
• m.padgett_at_physics.gla.ac.uk

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Moment of Inertia

• Moment of inertia of an object (I)
• rotational energy 1/2 I w2
• angular momentum I w
• Moment of inertia S miri2

m2
r2
m1
r1
r3
m3
3
Moment of inertia of a solid cylinder

• Moment of inertia of a thin ring

I Mr2 r2pr dr l r2
l

• Integrate to get a solid cylinder

R
I ?0R r2p l r3 dr 1/4 r2p l R4 1/2 mR2
r
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Moment of inertia of a hollow cylinder

• Moment of Inertia of a solid cylinder
• A hollow cylinder

I 1/2 mR2
-

m2
M
m1
R1
R2
I 1/2 m1R12 - 1/2 m2R22 1/2 M (R12 R22 )
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How fast does a cylinder roll?
Potential energy

• Equate energy at top of hill to energy at bottom

h
Kinetic energy rotational energy
E Mgh 1/2 Mv2 1/2 Iw2
potential
kinetic
rotational
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But this gets simpler

• Mgh 1/2 Mv2 1/2 Iw2
• Substitute for I 1/2 M (R12 R22)
• Mgh 1/2 Mv2 1/4 M (R12 R22) w2
• Substitute for w v/r1
• Mgh 1/2 Mv2 1/4 M (R12 R22)(v/R1)2

rotational
potential
kinetic
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.or (if not simpler) at least more useful!

• Note, mass cancels from each term
• gh 1/2 v2 1/4 (R12 R22)(v/R1)2
• Re-arrange for v
• v 2gh
• Note velocity depends NOT on mass, only on SHAPE
• As hole (R2) gets bigger hollow cylinder goes
  slower

1 1/2 (1 (R2 /R1 ) 2)
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Which cylinder rolls faster?

• The solid cylinder rolls faster

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Which material rolls faster?

• Both materials roll at the same speed