250
6 Oscillations
where I is the moment of inertia about its axis and C is known as torsional
constant given by τ = −Cθ , where τ is the torque.
6.37 In the arrangement shown in Fig. 6.10, the radius of the pulley is r , its moment
of inertia about the rotation axis is I and k is the spring constant. Assuming
that the mass of the thread and the spring is negligible and that the thread does
not slide over the frictionless pulley, calculate the angular frequency of small
oscillations.
Fig. 6.10
6.38 Two unstretched springs with spring constants k 1 and k 2 are attached to a solid
cylinder of mass m as in Fig. 6.11. When the cylinder is slightly displaced
and released it will perform small oscillations about the equilibrium position.
Assuming that the cylinder rolls without sliding, find the time period.
Fig. 6.11
6.39 A particle of mass m is located in a one-dimensional potential field U (x) =
a
x 2 −
b
x
where a and b are positive constants. Show that the period of small
oscillations that the particle performs about the equilibrium position will be
T = 4π
2a 3 m
b 4
[Osmania University 1999]