Question

6. (Bonus question.) A small uniform cylinder of radius R rolls without slipping along the inside of a large, fixed cylinder of radius r > R as shown in the figure above. mig (a) Use conservation of energy to show that the period of small oscillations (θ « 1) of the rolling cylinder about the equilibrium position O is equivalent to that of a simple pendulum whose length is (r-R). [Note that the rotation rate w of the cylinder is not equal to θ.] (b) Show the same result by considering forces and torques on the cylinder (relevant forces are shown above as an insert). 14 marks] Note: the equation of motion for a simple pendulum of length L in the small angle approx- imation is θ +(g/L)-0.

Answer 1

"Given that A small uniform cylinder of radius R roll without slipping along the inside of a large fixed cylinder of radius r greaterthan R. Total Energy of the cylinder for a given configuration E - mg (r - R) (1 - cos theta) + 1/2 mv^2 + 1/2 sw^2 mg (r - R) (1 - cos theta) + 1/2 mv^2 + 1/2 mR^2 /2 (V/R)^2 Lost steps belows there is rotating without stopping E - mg (r - R) (1 - cos theta) + 3/4 mv^2 Also, v = (r - R) theta \r\n Those E - mg (r - R) (1 - cos theta) + 3/2 mu (r - R^2) theta^2 d E/dt = 0 = mg (r - R) sin theta + 3/4 m (r - R^2) 2 theta theta = g (r - R) sin theta + 3/2 (r - R)^2 theta"" = 0 g sin theta + 3 (r - R) theta' = 0 For small theta, we have g theta + 3 (r - R) theta"" = 2 g^2 /3 (r - R) theta"