In the co-rotating frame, define an angular variable alpha such that the channel is at alpha = 0. You can enforce that the ball is at alpha = 0 using a Langrange multiplier, so you treat the alpha coordinate as a dynamic variable for the ball.
The kinetic energy is:
T = 1/2 m r-dot^2 + 1/2 m r^2 (omega + alpha-dot)^2
The Lagrangian is:
L = T - V - lambda alpha
where lambda is the lagrange multiplier that will make sure that the ball stays at alpha = 0. V is the potential energy, which is zero in this problem but you keep it in the equations as an unspecified function.
The derivatives of V would give you the force from the potential, which is zero, but this then allows you to put in the friction force by hand (friction force is, of course, not conservative and cannot be specified by a potential).
The value of lambda gives you the normal force from which you can compute the friction force. You can then add this friction force in the Euler Lagrange equations by hand to get the correct equation of motion.
A slotted disk is rotating with constant angular velocity ω−−, as shown in the figure. A ball of mass m slides out from the center of the disk. Kinetic friction force F=μk|N| acts on the mass, where N is the normal force between the ball and the channel. Using the body coordinate system i^, j^ fixed to the disk, derive the equation of motion for the mass using r(t) as the generalized coordinate.
Note: Gravity is not acting on this system.
Please enter symbolically the equation of motion in terms of r, r˙, m, ω, and μk.
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The answer is not clear! Can you explain ?
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