A light rope is attached to a block with mass 4.004.00 kg that rests on a frictionless, horizontal surface. The horizontal rope passes over a frictionless, massless pulley, and a block with mass mm is suspended from the other end. When the blocks are released, the tension in the rope is 15.015.0 N. How does the tension compare to the weight of the hanging block?

Ch 05: Applying Newton's Laws

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 05: Applying Newton's Laws

Problem 17d

Chapter 5, Problem 17d

A light rope is attached to a block with mass $4.00$ kg that rests on a frictionless, horizontal surface. The horizontal rope passes over a frictionless, massless pulley, and a block with mass $m$ is suspended from the other end. When the blocks are released, the tension in the rope is $15.0$ N. How does the tension compare to the weight of the hanging block?

Verified step by step guidance

Step 1: Begin by understanding the forces acting on the system. The tension in the rope (T = 15.0 N) is the same throughout the rope because the pulley is frictionless and massless. The hanging block experiences two forces: its weight (W = m * g) acting downward and the tension (T) acting upward.

Step 2: Recall that the weight of the hanging block is given by the formula:

W = m g

, where

g

is the acceleration due to gravity (approximately 9.8 m/s²). This weight is the force pulling the block downward.

Step 3: Compare the tension in the rope (T = 15.0 N) to the weight of the hanging block. If the tension is less than the weight, the block will accelerate downward. If the tension equals the weight, the block will remain stationary. If the tension is greater than the weight, the block will accelerate upward.

Step 4: To determine how the tension compares to the weight, calculate the weight of the hanging block using the formula

W = m g

. Substitute the value of

m

(mass of the hanging block) and

g

into the equation.

Step 5: Once the weight is calculated, compare it to the given tension (15.0 N). This comparison will reveal whether the tension is greater than, less than, or equal to the weight of the hanging block.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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 crucial for understanding the forces acting on both the block on the surface and the hanging block. It is the same throughout the rope in a frictionless system, meaning the tension experienced by the hanging block is equal to that experienced by the block on the surface.

Weight of an Object

The weight of an object is the force exerted on it due to gravity, calculated as the product of its mass and the acceleration due to gravity (W = mg). For the hanging block, its weight determines how much force is acting downward, which is essential for comparing it to the tension in the rope. In this case, the weight of the hanging block will influence whether it accelerates downward or remains at rest.

Newton's Second Law of Motion

Newton's Second Law states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass (F = ma). This principle helps analyze the forces acting on both blocks in the system. By applying this law, one can determine how the tension in the rope relates to the weight of the hanging block and the resulting motion of the system.

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Textbook Question

A light rope is attached to a block with mass $4.00$ kg that rests on a frictionless, horizontal surface. The horizontal rope passes over a frictionless, massless pulley, and a block with mass $m$ is suspended from the other end. When the blocks are released, the tension in the rope is $15.0$ N. What is the acceleration of either block?

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Textbook Question

A $750.0$ -kg boulder is raised from a quarry $125$ m deep by a long uniform chain having a mass of $575$ kg. This chain is of uniform strength, but at any point it can support a maximum tension no greater than $2.50$ times its weight without breaking. How long does it take to be lifted out at maximum acceleration if it started from rest?

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Textbook Question

A light rope is attached to a block with mass $4.00$ kg that rests on a frictionless, horizontal surface. The horizontal rope passes over a frictionless, massless pulley, and a block with mass m is suspended from the other end. When the blocks are released, the tension in the rope is $15.0$ N. Find $m$ .

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Textbook Question

When jumping straight up from a crouched position, an average person can reach a maximum height of about $60$ cm. During the jump, the person's body from the knees up typically rises a distance of around $50$ cm. To keep the calculations simple and yet get a reasonable result, assume that the entire body rises this much during the jump. With what initial speed does the person leave the ground to reach a height of $60$ cm?

A $550$ -N physics student stands on a bathroom scale in an elevator that is supported by a cable. The combined mass of student plus elevator is $850$ kg. As the elevator starts moving, the scale reads $450$ N. Find the acceleration of the elevator (magnitude and direction).

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