Static equilibrium and torque

Question

A board of mass 6.0 kg is connected to a hinge at one end. Here, L = 3.0m. A force F is applied vertically at the other end, and the board is tilted at 𝜃 = 30° with the horizontal. A 45-kg block rests on the board at d = .8m from the hinge.

Find the magnitude of the force F and find the force exerted by the hinge.

Answer 1

Generally, in these problems you do a torque balance around the hinge:

Clockwise torques: F_WBd(L/2)sin(90-θ) + F_WBlkd sin(90-θ)

CC torques: F(L)sin(90-θ)

You can equate these and solve for F upwards. Note that the angle between the lever arms and the forces is the complement of 30 degrees (theta). Note that board "location" is COM which is L/2

Note that you must put in mg = F_W for the board(Bd) and the Block(Blk)

The question is not physical in the sense that the upward force at the end is not generally purely upward. The usual thing would be to exert a perpendicular (to the board) force and take advantage of the hinge even more.

So, in this case there is now only a vertical force balance: F_H + F = F_W,Blk + F_W,Bd

Solve for F_H.

I hope that helped.

Answer 2

It can be easily solved if you include the diagram