- #1

ChiralSuperfields

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- 141

- Homework Statement
- Please seee below

- Relevant Equations
- Please seee below

FOr this,

Use alternate coordinate system

With ##ȳ##-axis parallel to incline and ##x̄##-axis parallel to the x-axis. Kinetic energy using this alternate coordinate system is ##T = \frac{1}{2}M\dot x_p^2 + \frac{1}{2}mR^2\dot \phi^2 + \frac{1}{2}m(\dot x̄^2 + \dot ȳ^2 + 2\dot x̄ \dot ȳ \cos\beta)## and potential energy is ##V = -mgȳ\sin \beta##. Note how potential energy of block was not included as we don’t know where the height of the COM is, however, it is a constant, so we can just all the high c, and it will add a ##+C## to the lagrangian

Now transform to original x-y coordinate system using transformations ##ȳ = -\frac{y}{sin \beta}## and ##x̄ = x + y\cot \beta## from decomposing ##ȳ## along the x and y axes. I think that ##\hat ȳ = \hat i - \hat j##. Then one would substitute the these transformations into the kinetic energy and potential energy expressions, however, how does one the ##x## and ##y## in these transformations in terms of ##x_p## and ##\phi##.

Thanks!

Use alternate coordinate system

With ##ȳ##-axis parallel to incline and ##x̄##-axis parallel to the x-axis. Kinetic energy using this alternate coordinate system is ##T = \frac{1}{2}M\dot x_p^2 + \frac{1}{2}mR^2\dot \phi^2 + \frac{1}{2}m(\dot x̄^2 + \dot ȳ^2 + 2\dot x̄ \dot ȳ \cos\beta)## and potential energy is ##V = -mgȳ\sin \beta##. Note how potential energy of block was not included as we don’t know where the height of the COM is, however, it is a constant, so we can just all the high c, and it will add a ##+C## to the lagrangian

Now transform to original x-y coordinate system using transformations ##ȳ = -\frac{y}{sin \beta}## and ##x̄ = x + y\cot \beta## from decomposing ##ȳ## along the x and y axes. I think that ##\hat ȳ = \hat i - \hat j##. Then one would substitute the these transformations into the kinetic energy and potential energy expressions, however, how does one the ##x## and ##y## in these transformations in terms of ##x_p## and ##\phi##.

Thanks!

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