At what angle does the upper end of the plank leave the wall?

[D_drayton]

If a uniform wooden plank is resting on a smooth floor reared vertically against a smooth wall. The plank's lower end is pushed gently to start it sliding freely away from the wall. As the plank slides its angle to the wall increases. At what angle does the upper end of the plank leave the wall?

Any ideas?

 

[arildno]

Welcome to PF!
Basically, you have 5 unknowns:
The 2 velocity components of C.M, the angular velocity of the plank about C.M., and the 2 normal forces, Nw and Ng, where Nw is the normal force acting from the wall, while Ng is the normal force acting from the ground.

Hence, you need 5 differential equations to solve this problem:
Newton's 2.law for C.M yields 2, the angular momentum equation yields 1,
and in addition, you must require that the normal velocity of the contact points on the
plank is zero. That's the remaining 2 equations.

You will get, I presume, a higly nonlinear system of equations, since, for example, the actual forces are unknown.
Good luck!

 

[M98Ranger]

The angle at which the upper end of the plank leaves the wall will depend on various factors such as the length and weight of the plank, the force applied to the lower end to start it sliding, and the coefficient of friction between the plank and the floor. It is difficult to determine an exact angle without this information. However, as the plank slides, the angle will continuously increase until the plank reaches its point of equilibrium where it will come to rest at a certain angle. This angle will depend on the factors mentioned above and can be calculated using trigonometry.