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 of the edge opposite the hinges.

검증된 단계별 안내

Identify the forces acting on the trapdoor: the weight of the trapdoor (300 N) acts downward at its center of gravity, and the upward force is applied either at the center or at the edge opposite the hinges.

For part (a), when the upward force is applied at the center, calculate the torque about the hinge. The torque due to the weight is \( \tau = \text{weight} \times \text{distance from hinge} \). Since the weight acts at the center, the distance is half the width of the door.

Set the net torque to zero to find the upward force needed to begin opening the door. The equation is \( F_{up} \times \frac{w}{2} = 300 \times \frac{w}{2} \), where \( F_{up} \) is the upward force and \( w \) is the width of the door.

For part (b), when the upward force is applied at the edge opposite the hinges, calculate the torque about the hinge again. The torque due to the weight remains the same, but the distance for the upward force is now the full width of the door.

Set the net torque to zero for this scenario: \( F_{up} \times w = 300 \times \frac{w}{2} \). Solve for \( F_{up} \) to find the force needed. Additionally, consider the forces at the hinge, which include both vertical and horizontal components due to the applied force and the weight of the door.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Torque

Torque is the rotational equivalent of force, defined as the product of force and the distance from the pivot point (lever arm). It determines how effectively a force can cause an object to rotate around an axis. In this problem, calculating torque helps determine the force needed to open the trapdoor by considering the distance from the hinge where the force is applied.

Center of Mass

The center of mass is the point in an object where its mass is evenly distributed and about which it balances in all directions. For a uniform object like the trapdoor, the center of mass is at its geometric center. Understanding this concept is crucial for calculating the torque when the force is applied at different points on the trapdoor.

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 counteracts the gravitational force and any additional forces exerted by the hinges.