Rotational Dynamics and angle of theta

[jman1211]

I tried to do this problem with related rates and calculus. We have not done physics with calculus, however. The other way I thought about was using the angular velocity of the board when the stick is taken away. I cannot seem to solve it though.

A common physics demonstration consists of a ball resting at the end of a board of length L that is elevated at an angle of theta with the horizontal. A light cup is attached to the board at Rc (Rc is a distance up the board from the bottom, it is not past the support stick) so that it will catch the ball when the support stick is suddenly removed. a) show that the ball will lag behind the falling board when theta < 35.3 and b) the ball will fall into the cup when the board is supported at this limiting angle and the cup is placed at
Rc = (2L)/(3cos(theta))

The board is hinged to the table.

Any hints would be wonderful.

 

[mukundpa]

Though I am not getting the problem clearly, I think you are missing the correct value of the velocity of the ball when it leaves the board, after that it is a projectile.

The potential energy lost is converted into the kinetic energy, both rotational and translational. the total kinetic energy of a ball having pure rotation is given by (7/10)mv^2 not (1/2)mv^2, where v is the velocity of center of mass.

May this help you !

 

[andrevdh]

The board is kept up at some angle with a stick. When it is removed the board falls flat, or hinges downwards. The cup therefore swings in beneath the ball which is falling straight downwards. Let me tell you what does not seem to work (I tried to solve it without using calculus). I assumed that the cm of the board falls downwards at g. Using it's y-coord and the distance it is mounted along the board one can calculate the angle of the board aa fn of time. This enables one to calculate the position of any point on he board wrt time. The difference between the ball's y-coord and the cup's y-coord did not get me to something that looks promising.