A uniform 300-N trapdoor in a floor is hinged at one side. Find the net upward force needed to begin to open it and the total force exerted on the door by the hinges (a) if the upward force is applied at the center and (b) if the upward force is applied at the center of the edge opposite the hinges.

Verified step by step guidance

1. Identify the forces acting on the trapdoor. These include the weight of the trapdoor acting downward through its center of mass and the upward force applied either at the center or at the center of the edge opposite the hinges.
2. Calculate the torque due to the weight of the trapdoor about the hinge. Torque (\(\tau\)) is given by the product of the force and the perpendicular distance from the point of rotation (hinge) to the line of action of the force. The weight acts at the center, so the distance is half the length of the trapdoor.
3. Calculate the torque due to the applied upward force. If the force is applied at the center, the distance from the hinge is again half the length of the trapdoor. If the force is applied at the center of the edge opposite the hinges, the distance is the full length of the trapdoor.
4. Set the sum of the torques equal to zero for equilibrium to find the magnitude of the upward force needed to just start opening the trapdoor. This involves solving the equation \(\tau_{weight} + \tau_{applied} = 0\).
5. Determine the total force exerted on the door by the hinges. This force must balance both the vertical forces (the weight and the upward force) and provide a counter-torque to keep the door from rotating.

Key Concepts

Torque

Torque is a measure of the rotational force applied to an object around a pivot point. It is calculated as the product of the force applied and the distance from the pivot point to the line of action of the force. In this scenario, understanding torque is essential to determine how the applied upward force affects the trapdoor's ability to rotate about its hinges.

Equilibrium

Equilibrium refers to a state where the sum of forces and the sum of torques acting on an object are both zero, resulting in no net movement. For the trapdoor, analyzing the forces acting on it, including the weight of the door and the applied upward force, is crucial to determine the conditions under which it will begin to open.

Force Distribution

Force distribution involves understanding how forces are transmitted through an object and how they affect different points, such as hinges. In this problem, the location of the applied force (center vs. edge) will influence the distribution of forces at the hinges, affecting the total force exerted on the door and its ability to rotate.