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b]1. Homework Statement [/b]
A cylinder (solid) of radius a rolls inside a fixed hollow cylinder of radius 4a; inside a homogenous gravity field.
Find, langragian using the rolling angle of the little cylidner as a generalized co-ord.
angular frequency of small oscilalations about equilibrium. is the the oscillation more or less rapid than that of a point particle sliding without friction inside the hollow cylinder?
L= T - V
[tex] T = 1/2 m \dot{r}^{2} + \frac{I}{2} \dot{\varphi}^{2} [/tex]
[tex] V = mgh [/tex]
[tex] chord length = 2Rsin(\frac{\theta}{2}) [/tex]
i have the lagrangian;
[tex] L = T-V = 1/2 M 12^{2} a^{2} \dot{\varphi}^{2} + 1/4 M a^{2} \dot{\varphi}^{2} - 3 \sqrt{2} a M g sin(2 \varphi) [/tex]
now i have trouble as when i got the small angle approximation [tex] sin(2 \varphi) [\tex] becomes [tex] 2 \varphi [/tex] and when i put this int the euler lagrange equation i end up differentiating this with respect to [tex] \varphi [/tex] and so i get a constant, then I am left with the [tex] \frac{d}{dt} \frac{\partial L}{\partial \dot{\varphi}}[/tex] bit being equal to a constant, [tex] 3 \sqrt{2} a m g [/tex]
which you can't solve with a function of cos and sine
A cylinder (solid) of radius a rolls inside a fixed hollow cylinder of radius 4a; inside a homogenous gravity field.
Find, langragian using the rolling angle of the little cylidner as a generalized co-ord.
angular frequency of small oscilalations about equilibrium. is the the oscillation more or less rapid than that of a point particle sliding without friction inside the hollow cylinder?
Homework Equations
L= T - V
[tex] T = 1/2 m \dot{r}^{2} + \frac{I}{2} \dot{\varphi}^{2} [/tex]
[tex] V = mgh [/tex]
[tex] chord length = 2Rsin(\frac{\theta}{2}) [/tex]
The Attempt at a Solution
i have the lagrangian;
[tex] L = T-V = 1/2 M 12^{2} a^{2} \dot{\varphi}^{2} + 1/4 M a^{2} \dot{\varphi}^{2} - 3 \sqrt{2} a M g sin(2 \varphi) [/tex]
now i have trouble as when i got the small angle approximation [tex] sin(2 \varphi) [\tex] becomes [tex] 2 \varphi [/tex] and when i put this int the euler lagrange equation i end up differentiating this with respect to [tex] \varphi [/tex] and so i get a constant, then I am left with the [tex] \frac{d}{dt} \frac{\partial L}{\partial \dot{\varphi}}[/tex] bit being equal to a constant, [tex] 3 \sqrt{2} a m g [/tex]
which you can't solve with a function of cos and sine
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