Two $25.0$ -N weights are suspended at opposite ends of a rope that passes over a light, frictionless pulley. The pulley is attached to a chain from the ceiling. What is the tension in the chain?

검증된 단계별 안내

Step 1: Begin by analyzing the forces acting on the pulley. The pulley is in equilibrium, meaning the net force acting on it is zero. The tension in the chain must balance the combined forces exerted by the two weights.

Step 2: Each weight exerts a force of 25.0 N downward due to gravity. Since the rope is frictionless and the pulley is light, the tension in the rope is the same on both sides of the pulley.

Step 3: The total downward force exerted on the pulley is the sum of the tensions from both sides of the rope. This is given by $ F_{total} = T_{rope} + T_{rope} $, where $ T_{rope} $ is the tension in the rope caused by each weight.

Step 4: Substitute the value of $ T_{rope} $ (which is equal to the weight of each object, 25.0 N) into the equation. The total downward force becomes $ F_{total} = 25.0 \text{ N} + 25.0 \text{ N} $.

Step 5: The tension in the chain must equal the total downward force to maintain equilibrium. Therefore, the tension in the chain is equal to $ F_{total} $.

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주요 개념

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

Tension in a Rope

Tension is the force transmitted through a rope or string when it is pulled tight by forces acting from opposite ends. In this scenario, the tension in the rope is equal to the weight of the objects suspended, as they are in equilibrium. Since both weights are equal, the tension in the rope will be the same as the weight of one of the objects.

Equilibrium

Equilibrium occurs when all the forces acting on an object are balanced, resulting in no net force and no acceleration. In this case, the two weights are equal and opposite, leading to a state of equilibrium. This principle allows us to analyze the forces in the system and determine the tension in the chain.

Pulley Systems

A pulley system consists of a wheel on an axle or shaft designed to support movement and change the direction of force. In this problem, the frictionless pulley allows the weights to move freely without resistance, simplifying the analysis of forces. The tension in the chain must support the total weight of the system, which is crucial for calculating the tension.