- #1

simon janssens

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## Homework Statement

The figure shows a uniform thin rigid plank of length 2b which can roll

without slipping on top of a rough circular log of radius a. The plank is initially

in equilibrium, resting symmetrically on top of the log, when it is slightly

disturbed. Find the period of small oscillations of the plank.

**2. Additional context**

The wording of the question implicates that we can use sinϑ ≈ ϑ and cosϑ ≈ 1 because ϑ

^{2}≈ 0.

The plank has no thickness and the mass is uniformly distributed.

Gravity is constant.

## The Attempt at a Solution

I figured that the resulting force is always pointed perpendicular to the plank in the center of mass. This results in a torque τ = r × F

_{⊥}.

r = |GC| = ϑa because there is no slipping

|F

_{⊥}| = mgcosϑ = mg (rounding for small angles)

so the torque is τ = mgaϑ

Then I try calculating the moment of inertia of the plank, I use the formula for parallel axes of rotation so that the moment of inertia is I = I

_{CM}+ mh

^{2}

I

_{CM}= ml

^{2}/12 = mb

^{2}/3

mh

^{2}= m|GC|

^{2}= ma

^{2}ϑ

^{2}= 0

so the moment of inertia is I = mb

^{2}/3

This gives me enough to calculate the angular acceleration, ϑ'' = τ/I = 3ga/b

^{2}⋅ ϑ

I deduce that ϑ is directly proportional with its second derivative.

ϑ has to be of the form e

^{λt}or Asin(ωt +φ

_{0}).

Because I'm looking for an oscillation, my guess is that it will be of the latter.

Now I have 3 variables and 3 equations, at t = 0 we get :

ϑ(0) = ϑ

_{0}= Asin(ω⋅0 +φ

_{0})

ϑ'(0) = 0 = ωAcos(ω⋅0 +φ

_{0})

ϑ''(0) = 3ga/b

^{2}⋅ ϑ

_{0}= -ω

^{2}Asin(ω⋅0 +φ

_{0})

I tried solving these, but it didn't work, can you help me please ?

Thanks a lot,

S.

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