Ch 05: Applying Newton's Laws

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 05: Applying Newton's Laws

Problem 17a

Chapter 5, Problem 17a

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. Draw two free-body diagrams: one for each block.

Verified step by step guidance

Step 1: Begin by identifying the forces acting on each block. For block A (on the horizontal surface), the forces are: (1) the tension in the rope pulling it to the right, and (2) the normal force and gravitational force acting vertically, which cancel each other out since the surface is frictionless. For block B (hanging vertically), the forces are: (1) the tension in the rope pulling it upward, and (2) the gravitational force pulling it downward.

Step 2: Draw the free-body diagram for block A. Represent block A as a box. Draw a horizontal arrow pointing to the right labeled 'T' (tension force). Draw a vertical arrow pointing downward labeled 'mg' (gravitational force), and a vertical arrow pointing upward labeled 'N' (normal force). Ensure the vertical forces are equal in magnitude and opposite in direction.

Step 3: Draw the free-body diagram for block B. Represent block B as a box. Draw a vertical arrow pointing upward labeled 'T' (tension force). Draw a vertical arrow pointing downward labeled 'mg' (gravitational force). These forces are not equal since the system is accelerating.

Step 4: Note that the tension in the rope is the same for both blocks because the rope is light and the pulley is frictionless. This means the tension force 'T' is consistent throughout the system.

Step 5: Use Newton's second law to analyze the forces. For block A, the net force is T = ma, where 'a' is the acceleration of the system. For block B, the net force is mg - T = ma. These equations can be used to solve for unknowns such as the acceleration or the mass of block B.

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Key Concepts

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

Free-Body Diagram

A free-body diagram is a graphical representation used to visualize the forces acting on an object. It shows all the external forces, including gravitational, normal, tension, and frictional forces, acting on the object in question. In this scenario, two free-body diagrams are needed: one for the block on the surface (Block A) and one for the hanging block (Block B), illustrating the forces each experiences.

Newton's Second Law

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, expressed as F = ma. This principle is crucial for analyzing the motion of the blocks in the system, as it allows us to relate the tension in the rope and the gravitational force acting on the hanging block to their respective accelerations.

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 problem, the tension in the rope is given as 15.0 N, which affects both blocks. Understanding how tension works in a pulley system is essential for determining the forces acting on each block and how they influence the overall motion of the system.

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Related Practice

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

An $8.00$ -kg block of ice, released from rest at the top of a $1.50$ -m-long frictionless ramp, slides downhill, reaching a speed of $2.50$ m/s at the bottom. What is the angle between the ramp and the horizontal?

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Textbook Question

An $8.00$ -kg block of ice, released from rest at the top of a $1.50$ -m-long frictionless ramp, slides downhill, reaching a speed of $2.50$ m/s at the bottom. What would be the speed of the ice at the bottom if the motion were opposed by a constant friction force of $10.0$ N parallel to the surface of the ramp?

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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

Three sleds are being pulled horizontally on frictionless horizontal ice using horizontal ropes (Fig. E $5.14$ ). The pull is of magnitude $190$ N. Find the tension in ropes $A$ and $B$ .

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Textbook Question

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