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 if the upward force is applied at the center.

검증된 단계별 안내

Identify the forces acting on the trapdoor. The weight of the trapdoor is 300 N, acting downward at its center of gravity, which is at the center of the trapdoor.

Determine the torque needed to begin opening the trapdoor. Torque (τ) is calculated using the formula: τ = r × F × sin(θ), where r is the distance from the hinge to the point where the force is applied, F is the force, and θ is the angle between the force and the lever arm. Since the force is applied vertically, θ = 90° and sin(θ) = 1.

For part (a), calculate the torque when the force is applied at the center of the trapdoor. The distance r is half the width of the trapdoor. Set the torque due to the applied force equal to the torque due to the weight of the trapdoor to find the required force.

For part (b), calculate the torque when the force is applied at the center of the edge opposite the hinges. The distance r is the full width of the trapdoor. Again, set the torque due to the applied force equal to the torque due to the weight of the trapdoor to find the required force.

Consider the forces exerted by the hinges. The hinges must support the weight of the trapdoor and provide a reaction force to balance the applied force. Use equilibrium conditions (sum of forces and sum of torques equal zero) to find the total force exerted by the hinges in each scenario.

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주요 개념

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Torque

Torque is a measure of the rotational force applied to an object. It is calculated as the product of the force and the distance from the pivot point (lever arm). In this problem, understanding torque is essential to determine the effect of the applied force on the trapdoor's rotation about the hinges.

Center of Mass

The center of mass is the point at which the mass of an object is considered to be concentrated. For a uniform object like the trapdoor, the center of mass is at its geometric center. This concept is crucial for calculating the gravitational torque acting on the door and understanding how the applied force affects its motion.

Equilibrium of Forces

Equilibrium occurs when all forces and torques acting on an object are balanced, resulting in no net motion. To find the net upward force needed to open the trapdoor, one must consider the equilibrium of forces and torques, ensuring that the applied force overcomes the gravitational torque to initiate rotation.