Question:
A solid cylinder is attached to a horizontal massless spring so that it can
roll without slipping
along a horizontal surface, as in Fig. 32.
The force constant k , of the spring is 
.
If the system is released from rest at a position in which the spring is
stretched by 
, find
, where M is the mass of the cylinder.
Solution:
We will start by converting everything to units we understand,
and 
.
Initially the cylinder is at rest and therefore the total mechanical energy of
the system is equal to the potential energy stored in the spring,
.
When the cylinder passes through its equilibrium position, the potential
energy stored in the spring vanishes, and therefore this total mechanical
energy is all in the form of kinetic energy, both translational and rotational.
In terms of the speed of the cylinder v, its mass, and its moment of inertia,
I, and angular speed 
, we have that
.
For a solid cylinder, we have that
, where R is the radius of the cylinder, we also have
that the angular speed is 
(no slipping).
Therefore, we have that
.
This leads to 
.
Hence we have that
as the cylinder passes through its equilibrium position, its translational
kinetic energy is
and its rotational energy is
.
To find the period, lets consider the forces acting on the cylinder.
The spring provides a force acting through the center of mass, 
, while
the contact with the surface provides a force 
at the surface of the
cylinder, and in the opposite direction to .
Newton's law tells us that the acceleration 
is related to the net force by
, while the angular acceleration is related to
the torque about the center of the cylinder
,
from which it follows that
.
Inserting this into Newton's law, we have that
gives
and hence
.
We recognize this as an equation describing simple harmonic motion,
, with
, from with we find a period of
.