Physics Problem Involving Trap Door Supported by Rope

Question

A uniform trapdoor shown below is 1.0 m by 1.5 m and weighs 300 N. It is supported by a single hinge (H), and by a light rope tied between the middle of the door and the floor. The door is held at the position shown, where its slab makes a 30 degree angle with the horizontal floor and the rope makes a 20 degree angle with the floor. Find the force at the hinge. Diagram

Answer 1

You have three unknowns: the tension in the string, and the vertical and horizontal components of the force at the hinge. So you need three equations. The first one is the torque equation with respect to the hinge. The only forces that have a torque are the weight and the tension in the string. Since you know all the angles, you have the tension.

Then you need to write the equilibrium for the forces. In the horizontal direction you have only the horizontal component of the force in the hinge and the horizontal component of the tension in the string, so they must be equal pointing in opposite direction. In the vertical direction you have the weight of the door, the vertical component of the tension in the string (both acting downward) and the vertical component in the hinge.