# Non-harmonic oscillatory motion

• Nitrus

#### Nitrus

I've got a test coming up with a problem similar to this one, I've figured out some of it but I am kinda lost on the rest, here it goes:
A solid sphere (radius = R) rolls without slipping in a cylindrical trough (radius = 5R). Show that, for small displacements from equilibrium perpendicular to the length of the trough, the sphere executes simple harmonic motion with a period T=2pi (28R/5g)^1/2.

Work:
I decided on taking an energy approach to this problem, and by doing so I must look at the KE of both the sphere and the effect the trough has on it.
$$v= \frac {ds} {dt} = 4R \frac {d\theta} {dt}$$
$$V=\frac {ds}{dt} = R\Omega$$
<p>
$$\Omega =\frac {V} {R} = 4 \frac {d\theta} {dt}$$
with that we have the following (also including moment of intertia for the sphere)
$$K = \frac {1} {2} 4R {\frac {d\theta}{dt}}^2 + \frac {1} {2}(\frac{2} {5} mR^2)(4{\frac {d\theta}{dt}}^2))$$
the trough is a half circle by the way...
that all simplifies to
$$((\frac {d\theta}{dt}))^2 \frac {56mR^2}{5}$$
so now i have the energy of the system, what should i do next?

Last edited by a moderator:

$$4mgR(1-\cos\theta)\approx 2mgR\theta^2$$, note that energy is conserved, try harmonic variation for $$\theta$$, and you'll find your answer.