How to solve the problem?

Question

A bench that is 3.5 m long has a mass of 100 kg. It is supported at the left end by a wooden post. The right post is 2.0 m from the right end of the bench. A 200 N person sits 1.0 m from the right end. The magnitude of the force exerted by the right wooden post is _____N. The magnitude of the force exerted by the left wooden post is ____ N.

Answer 1

If the system is in mechanical equilibrium, then both the sum of forces and the sum of torques (wrt to the locations where the bench is supported) must be 0. Torque (wrt a point) is computed by multiplying the force by the distance between the reference point and the point it is applied to.

If A denotes the left end of the bench, where the left post is, there are three torque contributions: the weight of the bench (torque = 100kg * 10m/s^2 * 3.5m/2, clockwise), the weight of the person (torque = 200N * (3.5-1)m, also clockwise) and the force exerted by the right post (torque = F_B * (3.5-2)m, counterclockwise). Note that the force exerted by the both posts are unknown, and that F_A does not contribute torque.

By balancing the torques, we can find F_B.

1750Nm + 500Nm = F_B * 1.5m. ⇒. F_B = 1500 N

Similarly, we can compute the torques wrt the right post, to determine F_A. Or, alternatively, we can use balance of forces: besides the weight or the person and the bench, there are F_A and F_B, and the sum needs to be 0, and we can solve for F_A = -300N

Please note the F_A is negative, but F_B is positive. This means the post on the left "pulls" down the bench, while the post on the right "pushes".