 56
 0
1. The problem statement, all variables and given/known data
We have a ball of mass ##m##and radius ##r##. it is placed on an incline (We don't know the angle of the incline, nor we do whether the angle is constant along the incline  maybe it is a curved incline) and then released. The COM of ball is ##h## meters above the incline at the initial time (##r## is comparable to ##h## and is not negligible). Incline has mass ##M## which is also comparable to ##m## and is not very large, so incline will also move. We also have ##\mu_s## and ##\mu_k##, static and kinetic friction coefficient. when the ball reaches the horizontal part of the incline, we want to find ##v## and ##u##, velocity of ball and incline, relative to ground. (##I## moment of inertia of Ball is known. ##I= 2/3 mr^2##)
(As you can see, the incline is somehow curved and the angle of slope is not constant)
3. The attempt at a solution
I can solve these type of question for a completely not moving incline with angle ##\theta##. I write ##\tau = I \alpha## and ##F = m a## and ##a = \alpha r## and by knowing that at the first ##\tau = r ~mg~cos\theta ~~\mu_k##, everything is solved easily. And also we can find a point where ##v = \omega r##.
But In this question, we don't know anything about the incline except that it has a Mass and moves. And I don't know how to approach this problem.
We have a ball of mass ##m##and radius ##r##. it is placed on an incline (We don't know the angle of the incline, nor we do whether the angle is constant along the incline  maybe it is a curved incline) and then released. The COM of ball is ##h## meters above the incline at the initial time (##r## is comparable to ##h## and is not negligible). Incline has mass ##M## which is also comparable to ##m## and is not very large, so incline will also move. We also have ##\mu_s## and ##\mu_k##, static and kinetic friction coefficient. when the ball reaches the horizontal part of the incline, we want to find ##v## and ##u##, velocity of ball and incline, relative to ground. (##I## moment of inertia of Ball is known. ##I= 2/3 mr^2##)
(As you can see, the incline is somehow curved and the angle of slope is not constant)
3. The attempt at a solution
I can solve these type of question for a completely not moving incline with angle ##\theta##. I write ##\tau = I \alpha## and ##F = m a## and ##a = \alpha r## and by knowing that at the first ##\tau = r ~mg~cos\theta ~~\mu_k##, everything is solved easily. And also we can find a point where ##v = \omega r##.
But In this question, we don't know anything about the incline except that it has a Mass and moves. And I don't know how to approach this problem.
Attachments

103.3 KB Views: 83