What force perpendicular to hinge is needed to support a gate?

Question

A uniform 45 pound gate, 3 ft. wide and 5 ft. high is supported by two hinges. Each hinge is located 6 in. from the top and bottom of the gate. What force, in pounds, acting perpendicular to the upper hinge must the upper hinge supply to support the gate? I have been working on this problem with no success and it's due tomorrow! Help! Please explain fully how to arrive at the answer.

Answer 1

Think torque.

Since the gate isn't moving, the net torque around any axis must be zero.

Take the lower hinge as your axis. The two torques are due to the force on the upper hinge and the force of gravity on the gate. They must balance.

Recall that torque = force x radial distance

The radius for the gate is from the center of mass for the gate. The radius for the upper hinge is the distance between the two hinges.

The force of gravity is down. The force from the upper hinge has components perpendicular to the radial line (i.e. the line to the lower hinge) and components parallel to it. Only the latter count for the torque.