Ch 07: Newton's Third Law

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 07: Newton's Third Law

Problem 24

Chapter 7, Problem 24

The 100 kg block in FIGURE EX7.24 takes 6.0 s to reach the floor after being released from rest. What is the mass of the block on the left? The pulley is massless and frictionless.

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Identify the forces acting on the system: The 100 kg block on the right experiences a downward gravitational force $ F_{g, \text{right}} = m_{\text{right}} g $, where $ m_{\text{right}} = 100 \ \text{kg} $ and $ g = 9.8 \ \text{m/s}^2 $. The block on the left (mass $ m_{\text{left}} $) experiences a downward gravitational force $ F_{g, \text{left}} = m_{\text{left}} g $. The tension in the rope is the same throughout because the pulley is massless and frictionless.

Write the equations of motion for each block: For the 100 kg block (right block), the net force is $ F_{\text{net, right}} = m_{\text{right}} a = m_{\text{right}} g - T $, where $ T $ is the tension in the rope. For the left block, the net force is $ F_{\text{net, left}} = m_{\text{left}} a = T - m_{\text{left}} g $. Here, $ a $ is the acceleration of the system, which is the same for both blocks.

Relate the acceleration to the motion: The 100 kg block takes 6.0 s to reach the floor. Using the kinematic equation $ d = \frac{1}{2} a t^2 $, where $ d $ is the distance the block falls, $ t = 6.0 \ \text{s} $, and $ a $ is the acceleration, solve for $ a $.

Combine the equations of motion: Substitute $ a $ from the kinematic equation into the equations of motion for both blocks. Add the two equations to eliminate $ T $, resulting in $ m_{\text{right}} g - m_{\text{left}} g = (m_{\text{right}} + m_{\text{left}}) a $. Solve for $ m_{\text{left}} $ in terms of known quantities.

Substitute the known values: Use $ m_{\text{right}} = 100 \ \text{kg} $, $ g = 9.8 \ \text{m/s}^2 $, and the calculated $ a $ from the kinematic equation to find $ m_{\text{left}} $.

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Key Concepts

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

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. This principle is crucial for analyzing the forces acting on the blocks and the resulting motion, allowing us to calculate the acceleration and the forces involved in the system.

Free Fall and Acceleration due to Gravity

Free fall refers to the motion of an object under the influence of gravity alone, with an acceleration of approximately 9.81 m/s² near the Earth's surface. Understanding free fall is essential for determining how long it takes for the block to reach the ground and how this relates to the forces acting on the other block in the pulley system.

Pulley Systems and Tension

In a pulley system, tension is the force transmitted through the rope or cable, which affects the motion of the connected masses. Since the pulley is massless and frictionless, the tension is the same throughout the rope, allowing us to set up equations that relate the masses and their accelerations, ultimately leading to the solution for the unknown mass.

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Textbook Question

A mobile at the art museum has a 2.0 kg steel cat and a 4.0 kg steel dog suspended from a lightweight cable, as shown in FIGURE EX7.21. It is found that $theta sub 1$ = 20° when the center rope is adjusted to be perfectly horizontal. What are the tension and the angle of rope 3?

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Textbook Question

An 85 kg cheerleader stands on a scale that reads in kg. What does the scale read if the 85 kg cheerleader lifts the 50 kg cheerleader upward with an acceleration of 2.0 m/s²?

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A 500 kg air conditioner sits on the flat roof of a building. The coefficient of static friction between the roof and the air conditioner is 0.90. A massless rope attached to the air conditioner passes over a massless, frictionless pulley at the edge of the roof. In an effort to drag the air conditioner to the edge of the roof, four 100 kg students hang from the free end of the rope, but the air conditioner refuses to budge. What is the magnitude of the rope tension at the point where it is attached to the air conditioner?

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Textbook Question

Two blocks are attached to opposite ends of a massless rope that goes over a massless, frictionless, stationary pulley. One of the blocks, with a mass of 6.0 kg, accelerates downward at (3/4)g. What is the mass of the other block?

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Textbook Question

The 1.0 kg block in FIGURE EX7.23 is tied to the wall with a rope. It sits on top of the 2.0 kg block. The lower block is pulled to the right with a tension force of 20 N. The coefficient of kinetic friction at both the lower and upper surfaces of the 2.0 kg block is μ_k = 0.40. What is the tension in the rope attached to the wall?

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Textbook Question

A rope of length L and mass m is suspended from the ceiling. Find an expression for the tension in the rope at position y, measured upward from the free end of the rope.

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