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Angelo Niforatos
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Hi all! I am working on finding the Lagrangian for the situation stated in the title. This is actually a Wolfram Mathematica demonstration as well in which they give you the Lagrangian. I am working on re-deriving it.
I seem to be missing four terms in my Lagrangina that I believe stems from my equation for Kinetic energy of the disk and ring. The answer can be found in the link provided. http://demonstrations.wolfram.com/DiskRollingInsideARotatingRing/
My attempt at the solution is below.
Position of disk: $$ r(t) = (R-r)\sin\theta \hat{x} - (R-r)\cos\theta \hat{y} $$
Velocity of Disk $$\dot{r} = (R-r)\dot{\theta}\cos\theta \hat{x} + (R-r)\dot{\theta}\sin\theta \hat{y}$$
Velocity of Ring $$\dot{r} = -R\dot{\phi}\sin\phi \hat{x} + R\dot{\phi}\cos\phi \hat{y}$$
Total Kinetic Energy for ring and disk: T = $$\frac{1}{2}I_{ring}\dot{\phi}^2 + \frac{1}{2}I_{disk}\dot{\alpha}^2 + \frac{1}{2}m_{disk}((R-r)^2\dot{\theta}^2) $$
Where alpha is given in the Wolfram Demonstartion that basically just says that since friction is sufficient enough for no slipping the disk should rotate with the ring.
Potential Energy: $$ U = -m_{disk}g(R-r)\cos\theta $$
The four terms I am missing are $$ \frac{1}{2}m_2(2Rr\dot{\theta}\dot{\phi}\cos\theta - 2R^2\dot{\theta}\dot{\phi}\cos\theta + I_{ring}R^2\dot{\phi}^2 + m_{ring}r^2R^2\dot{\phi}^2) $$
Or, $$m_{disk}R\dot{\phi}\dot{\theta}( (r-R)\cos\theta) + R^2\dot{\phi}^2(I_{ring} + m_{ring}r^2)$$
Thank you!
Hi all! I am working on finding the Lagrangian for the situation stated in the title. This is actually a Wolfram Mathematica demonstration as well in which they give you the Lagrangian. I am working on re-deriving it.
I seem to be missing four terms in my Lagrangina that I believe stems from my equation for Kinetic energy of the disk and ring. The answer can be found in the link provided. http://demonstrations.wolfram.com/DiskRollingInsideARotatingRing/
My attempt at the solution is below.
Position of disk: $$ r(t) = (R-r)\sin\theta \hat{x} - (R-r)\cos\theta \hat{y} $$
Velocity of Disk $$\dot{r} = (R-r)\dot{\theta}\cos\theta \hat{x} + (R-r)\dot{\theta}\sin\theta \hat{y}$$
Velocity of Ring $$\dot{r} = -R\dot{\phi}\sin\phi \hat{x} + R\dot{\phi}\cos\phi \hat{y}$$
Total Kinetic Energy for ring and disk: T = $$\frac{1}{2}I_{ring}\dot{\phi}^2 + \frac{1}{2}I_{disk}\dot{\alpha}^2 + \frac{1}{2}m_{disk}((R-r)^2\dot{\theta}^2) $$
Where alpha is given in the Wolfram Demonstartion that basically just says that since friction is sufficient enough for no slipping the disk should rotate with the ring.
Potential Energy: $$ U = -m_{disk}g(R-r)\cos\theta $$
The four terms I am missing are $$ \frac{1}{2}m_2(2Rr\dot{\theta}\dot{\phi}\cos\theta - 2R^2\dot{\theta}\dot{\phi}\cos\theta + I_{ring}R^2\dot{\phi}^2 + m_{ring}r^2R^2\dot{\phi}^2) $$
Or, $$m_{disk}R\dot{\phi}\dot{\theta}( (r-R)\cos\theta) + R^2\dot{\phi}^2(I_{ring} + m_{ring}r^2)$$
Thank you!
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