- #1
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.
[tex]
v= \frac {ds} {dt} = 4R \frac {d\theta} {dt}
[/tex]
[tex]
V=\frac {ds}{dt} = R\Omega
[/tex]
<p>
[tex]
\Omega =\frac {V} {R} = 4 \frac {d\theta} {dt}
[/tex]
with that we have the following (also including moment of intertia for the sphere)
[tex]
K = \frac {1} {2} 4R {\frac {d\theta}{dt}}^2 + \frac {1} {2}(\frac{2} {5} mR^2)(4{\frac {d\theta}{dt}}^2))
[/tex]
the trough is a half circle by the way...
that all simplifies to
[tex]
((\frac {d\theta}{dt}))^2 \frac {56mR^2}{5}
[/tex]
so now i have the energy of the system, what should i do next?
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.
[tex]
v= \frac {ds} {dt} = 4R \frac {d\theta} {dt}
[/tex]
[tex]
V=\frac {ds}{dt} = R\Omega
[/tex]
<p>
[tex]
\Omega =\frac {V} {R} = 4 \frac {d\theta} {dt}
[/tex]
with that we have the following (also including moment of intertia for the sphere)
[tex]
K = \frac {1} {2} 4R {\frac {d\theta}{dt}}^2 + \frac {1} {2}(\frac{2} {5} mR^2)(4{\frac {d\theta}{dt}}^2))
[/tex]
the trough is a half circle by the way...
that all simplifies to
[tex]
((\frac {d\theta}{dt}))^2 \frac {56mR^2}{5}
[/tex]
so now i have the energy of the system, what should i do next?
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