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Chapter 5: Problem 1

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?

Short Answer

Expert verified
Tension in the rope: 25.0 N; Tension in the chain: 50.0 N.

Step by step solution

01

Understanding the System

We have a system with two weights, each weighing 25.0 N, suspended from the ends of a rope over a frictionless pulley. The key point to note is that the pulley does not add any additional forces, given that it is frictionless and light.
02

Analyzing Forces on the Weights

Each weight is 25.0 N, which means the force due to gravity acting on each of them is 25.0 N downward. Because the system is in balance (the weights are equal and the system is static), the tension in the rope must be equal to the force exerted by each weight, which is 25.0 N.
03

Calculating the Tension in the Rope

Since the system is symmetric and in equilibrium, the tension in the rope between the pulley and each weight must equal the weight's force. Thus, the tension in the rope, T, is 25.0 N.
04

Analyzing the Pulley's Equilibrium

For the pulley to remain in equilibrium, the tensions in the sections of the rope on either side of the pulley must provide an upward force equal to the sum of the forces from the weights. Since each side contributes a tension of 25.0 N to the pulley, the total upward tension in the chain attached to the pulley is the sum of these two tensions.
05

Calculating the Tension in the Chain

The tension in the chain is the sum of the tensions from both sides of the rope (both 25.0 N), which equals 50.0 N, counteracting the total downward force of both suspended weights.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Tension in Ropes
Tension in ropes plays a crucial role in physics, especially in systems involving weights and pulleys. When a rope is pulled by forces at both ends, the tension is the internal force along the rope that resists these external pulls. In our example, each weight creates a force of 25.0 N due to gravity.
In such a system, equilibrium is achieved, and the tension throughout the rope remains constant. It's essential to realize that in an ideal scenario where the pulley is frictionless and the rope has no mass, the tension in the rope equals the weight pulling on it. This gives us a clear and simple understanding: the tension in the rope between the weights and the pulley remains at 25.0 N.
Understanding tension helps solve complex real-world problems involving cables, ropes, and pulleys in engineering and construction projects.
Equilibrium in Physics
Equilibrium denotes a state where a system experiences no net force, enabling it to be at rest or move with constant velocity. In physics, especially Newtonian mechanics, achieving equilibrium requires balancing forces to sum to zero.
In the provided exercise, equilibrium manifests through each 25.0 N force exerted by the weights being perfectly balanced by the tension in the rope. For the pulley itself, equilibrium is achieved when the sum of the tensions counteracts the total weight suspended. Hence, while each side of the rope contributes 25.0 N upward, the chain holding the pulley must exert a total force of 50.0 N upwards to maintain equilibrium.
Understanding equilibrium is crucial in mechanics, as it lays the foundation for analyzing forces in static and dynamic systems.
Free-Body Diagrams
Free-body diagrams are valuable tools for visualizing forces acting on a body. They help break down complex relationships into simpler parts. In our scenario, drawing a free-body diagram involves identifying forces like tension and gravity acting on the weights and pulley.
The weights each have downward force vectors representing their weight (25.0 N each). Meanwhile, the tension in the rope, acting upwards towards the pulley, is equal to 25.0 N on each side, shown by upward force vectors. For the pulley, a free-body diagram highlights the upward forces (tensions) on either side, balancing the downward forces, exerted by the weights, making the system static.
By mastering free-body diagrams, students distinguish between and understand individual forces, leading to improved problem-solving in physics.

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Most popular questions from this chapter

A 50.0-kg stunt pilot who has been diving her airplane vertically pulls out of the dive by changing her course to a circle in a vertical plane. (a) If the plane's speed at the lowest point of the circle is 95.0 \(\mathrm{m} / \mathrm{s}\) , what is the minimum radius of the circle for the acceleration at this point not to exceed 4.00 \(\mathrm{g} ?\) (b) What is the apparent weight of the pilot at the lowest point of the pullout?

An \(85-N\) box of oranges is being pushed across a horizontal floor. As it moves, it is slowing at a constant rate of 0.90 \(\mathrm{m} / \mathrm{s}\) each second. The push force has a horizontal component of 20 \(\mathrm{N}\) and a vertical component of 25 \(\mathrm{N}\) downward. Calculate the coefficient of kinetic friction between the box and floor.

On the ride "Spindletop" at the amusement park Six Flags Over Texas, people stood against the inner wall of a hollow vertical cylinder with radius 2.5 \(\mathrm{m}\) . The cylinder started to rotate, and when it reached a constant rotation rate of 0.60 rev \(/ \mathrm{s}\) , the floor on which people were standing dropped about 0.5 \(\mathrm{m}\) . The people remained pinned against the wall. (a) Draw a force diagram for a person on this ride, after the floor has dropped. (b) What minimum coefficient of station is required if the person on the ride is not to slide downward to the new position of the floor? (c) Does your answer in part (b) depend on the mass of the passenger? (Note: When the ride is over, the cylinder is slowly brought to rest. As it slows down, people side down the walls to the floor.)

The Cosmoclock 21 Ferris wheel in Yokohama City, Japan, has a diameter of 100 \(\mathrm{m}\) . Its name comes from its 60 arms, each of which can function as a second hand (so that it makes one revolution every 60.0 \(\mathrm{s}\) ). (a) Find the speed of the passengers when the Ferris wheel is rotating at this rate. (b) A passenger weighs 882 \(\mathrm{N}\) at the weight-guessing booth on the ground. What is his apparent weight at the highest and at the lowest point on the Ferris wheel? (c) What would be the time for one revolution if the passenger's apparent weight at the highest point were zero? (d) What then would be the passenger's apparent weight at the lowest point?

Block \(B\) , with mass 5.00 \(\mathrm{kg}\) , rests on block \(A\) , with mass 8.00 \(\mathrm{kg}\) , which in turn is on a horizontal tabletop (Fig. 5.72 ). There is no friction between block \(A\) and the tabletop, but the coefficient of static friction between block \(A\) and block \(B\) is \(0.750 .\) A light string attached to block \(A\) passes over a frictionless, massless pulley, and block \(C\) is suspended from the other end of the string. What is the largest mass that block \(C\) can have so that blocks \(A\) and \(B\) still slide together when the system is released from rest? figure can't copy
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