1. The problem statement, all variables and given/known data
1) A mouse of mass ##m## runs around the inner circumference of a vertical circle which is free to rotate about its centre. The mouse starts at the bottom of a stationary wheel. Let ##\phi## be the angle that the radius vector of the mouse makes with the downward vertical at time ##t## and let the distance run by the mouse around the circumference at time ##t## be a known function ##s(t)##. Show that the Lagrangian for ##\phi(t)## is $$L = \frac{1}{2}ma^2\dot{\phi}^2 + \frac{1}{2}I\left(\frac{\dot{s}}{a}-\dot{\phi}\right) + mga\cos \phi$$
(only need help with this part - but the rest of the question is attached for clarity (AttachmentQ2))
2)In attachment Q1 Last part: If the particle has speed ##u## when passing through O, show that its speed relative to the wire when passing through the point P at the other end of the diameter is ##\sqrt{u^2 + 4a^2 w^2}##. (Rest of question provided for clarity again)
2. Relevant equations
Kinetic energies, angular velocity.
3. The attempt at a solution
1)The first term in the Lagrangian is the translational kinetic energy of the mouse relative to an inertial frame and the third term is its potential energy. Introducing ##\theta## as the angle through which a point on the surface of the wheel has rotated, assuming there is no slip between the mouse and the inner circumference, then ##\dot{\theta} = \dot{s}/a##. Let ##w## be the angular velocity of the mouse wrt an inertial frame. How do we obtain ##w = \dot{s}/a - \dot{\phi}##? Why isn't ##\dot{\phi}## enough?
2) I think to get the speed at P relative to the wire then I need to solve ##v_p = 2a \dot{\phi}##. From a previous part of the question, I got that the equation of motion for ##\phi## incorporating the constraint ##\dot{\theta}=w##, is $$\ddot{\phi} + w^2 \sin \phi = 0$$ I could multiply this equation by ##\dot{\phi}##, integrate to get d/dt (expression) = constant and use the initial conditions for the velocity at O to get the constant. This would give me ##\phi## but the integral was non-elementary. Even so, if I could solve it, it would give me ##\phi(t)## and I would not know the time taken for the particle to get to P.
Many thanks.
1) A mouse of mass ##m## runs around the inner circumference of a vertical circle which is free to rotate about its centre. The mouse starts at the bottom of a stationary wheel. Let ##\phi## be the angle that the radius vector of the mouse makes with the downward vertical at time ##t## and let the distance run by the mouse around the circumference at time ##t## be a known function ##s(t)##. Show that the Lagrangian for ##\phi(t)## is $$L = \frac{1}{2}ma^2\dot{\phi}^2 + \frac{1}{2}I\left(\frac{\dot{s}}{a}-\dot{\phi}\right) + mga\cos \phi$$
(only need help with this part - but the rest of the question is attached for clarity (AttachmentQ2))
2)In attachment Q1 Last part: If the particle has speed ##u## when passing through O, show that its speed relative to the wire when passing through the point P at the other end of the diameter is ##\sqrt{u^2 + 4a^2 w^2}##. (Rest of question provided for clarity again)
2. Relevant equations
Kinetic energies, angular velocity.
3. The attempt at a solution
1)The first term in the Lagrangian is the translational kinetic energy of the mouse relative to an inertial frame and the third term is its potential energy. Introducing ##\theta## as the angle through which a point on the surface of the wheel has rotated, assuming there is no slip between the mouse and the inner circumference, then ##\dot{\theta} = \dot{s}/a##. Let ##w## be the angular velocity of the mouse wrt an inertial frame. How do we obtain ##w = \dot{s}/a - \dot{\phi}##? Why isn't ##\dot{\phi}## enough?
2) I think to get the speed at P relative to the wire then I need to solve ##v_p = 2a \dot{\phi}##. From a previous part of the question, I got that the equation of motion for ##\phi## incorporating the constraint ##\dot{\theta}=w##, is $$\ddot{\phi} + w^2 \sin \phi = 0$$ I could multiply this equation by ##\dot{\phi}##, integrate to get d/dt (expression) = constant and use the initial conditions for the velocity at O to get the constant. This would give me ##\phi## but the integral was non-elementary. Even so, if I could solve it, it would give me ##\phi(t)## and I would not know the time taken for the particle to get to P.
Many thanks.
0 commentaires:
Enregistrer un commentaire