Problem 1 - University Physics with Modern Physics

Question

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 that goes to the ceiling. (a) What is the tension in the rope? (b) What is the tension in the chain?

Step-by-Step Solution

Answer

(a) 25.0 N; (b) 50.0 N.

Step 1: Understanding the System

In this problem, we have a system where two 25.0 N weights are hanging on opposite sides of a rope over a pulley. The system is symmetrical — each weight pulls down on the rope with a force equal to their weight.

Step 2: Analyzing Forces on the Rope

The system is in equilibrium because the weights are equal and opposite. The tension in any rope section is equal to the force exerted by each weight. Since both weights are equal to 25.0 N, the tension in the rope is also 25.0 N.

Step 3: Calculating Tension in the Rope

Since the tension throughout the rope must be the same, and both weights are pulling equally, the tension in the rope is directly equal to the weight of either weight. Thus, the tension in the rope is 25.0 N.

Step 4: Understanding Tension in the Chain

The chain supports the pulley and the weights. For the system to be in equilibrium, the chain must provide enough force to balance the total downward force exerted by the weights. This means it must compensate for both weights.

Step 5: Calculating Total Force on the Chain

The total tension in the chain is the sum of the forces from both weights, which is 25.0 N from each weight. Therefore, the total force exerted on the chain from the weights is 50.0 N.

Key Concepts

Equilibrium in Physics

When we talk about equilibrium in physics, we're referring to a state where all forces acting on a system are perfectly balanced. In simpler terms, equilibrium is the balance point that keeps a system steady and unmoving. In our pulley system problem, the key is that each 25.0 N weight pulls equally on the rope, creating a state of equilibrium. For equilibrium to be achieved, the forces must be equal in size but opposite in direction, ensuring the system doesn't unintentionally accelerate or move. Without this balance, the weights in the pulley system would tip in one direction or the other, disrupting equilibrium and causing motion.

Force Balance

Force balance is crucial in understanding how systems like our pulley problem stay still. Imagine pulling on either side of a rope with equal but opposite forces. These forces cancel each other out, maintaining a stationary position. This is exactly what's happening in our exercise. The 25.0 N weights each pull down on the rope, yet because these forces are equal and opposing, the rope doesn't move.

• The force from the first weight is 25.0 N downward.
• The force from the second weight is also 25.0 N downward.

Because these forces are balanced, the tension felt throughout the rope is the same at every point. This force balance concept explains why the rope tension matches the force from a single weight: 25.0 N.

Pulley Systems

Pulley systems, like in our exercise, offer a fantastic way to understand how force is distributed and balanced. A pulley consists of a wheel around which a rope can move, changing the direction of the force exerted on it. In our problem, the pulley is frictionless and light, which means it doesn't affect the tension calculations directly. Instead, it simply changes the direction of the forces acting on the weights.

When running through these kinds of problems:

• Recognize that a single rope in the system means the tension is the same throughout.
• The chain attached to the ceiling supports the combined weight, thus the tension in the chain adds up to 50.0 N, balancing both hanging weights.
• In real-world applications, understanding these mechanics helps in designing efficient systems for lifting or holding weights with minimal effort.

Pulley systems effectively distribute forces, minimizing the required output to achieve balance.