Solve Rotation Q: Period of Small Oscillations for Rolling Cylinder

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SUMMARY

The discussion focuses on deriving the period of small oscillations for a small uniform cylinder rolling inside a larger fixed cylinder. The conclusion is that the period is equivalent to that of a simple pendulum with a length of 3(b - a)/2, where b is the radius of the larger cylinder and a is the radius of the smaller cylinder. Key equations include the rotational kinetic energy expressed as 0.5Iw² and the gravitational potential energy as mg(1 - cos(p)). The relationship between angular velocity and the angle p is critical for solving the problem.

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  • Understanding of rotational dynamics and kinetic energy
  • Familiarity with gravitational potential energy concepts
  • Knowledge of simple harmonic motion and pendulum mechanics
  • Basic trigonometry, particularly involving angles and cosine functions
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  • Study the derivation of the period of a simple pendulum
  • Explore the relationship between angular velocity and linear velocity in rolling motion
  • Investigate energy conservation principles in rotational systems
  • Learn about the dynamics of rolling without slipping conditions
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Physics students, mechanical engineers, and anyone interested in the dynamics of rolling motion and oscillatory systems will benefit from this discussion.

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Homework Statement


A small uniform cylinder of radius a rolls without slipping on the inside of a large fixed cylinder of radius b (b>=a). Show that the period of small oscillations of the rolling cylinder is that of a simple pendulum of length 3(b - a)/2


Homework Equations


rotational KE = 0.5Iw^2


The Attempt at a Solution



OK I tried to do this with energy considerations. I called the angle between the vertical and the line from the centre of the big cyclinder to the position of the small one p, and so let the GPE be mg(1 - cosp) and expanded cosp to 2 terms. I added this to the KE of the centre of mass and the rotational energy 0.5Iw^2 of the cylinder.

I therefore need some sort of relationship between w, the angular velocity of the cylinder about it's centre, and the angle p. I thought perhaps adp/t=bw, as the point on the cylinder in contact with the big cylinder will have a tangential velocity described by both. But basically I'm stumped - please help!
 
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