In FIGURE CP7.54, find an expression for the acceleration of m1. The pulleys are massless and frictionless. Hint: Think carefully about the acceleration constraint.

Ch 07: Newton's Third Law

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 07: Newton's Third Law

Problem 54

Chapter 7, Problem 54

In FIGURE CP7.54, find an expression for the acceleration of m₁. The pulleys are massless and frictionless. Hint: Think carefully about the acceleration constraint.

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Identify the forces acting on each mass. For m₁, the forces are its weight (m₁g) acting downward and the tension (T) in the string acting upward. For m₂, the forces are its weight (m₂g) acting downward and the tension (T) in the string acting upward. Assume the pulleys are massless and frictionless, so the tension is the same throughout the string.

Establish the acceleration constraint. Since the string is inextensible, the accelerations of m₁ and m₂ are related. If m₁ moves upward with acceleration a₁, m₂ must move downward with acceleration a₂. The relationship between these accelerations depends on the pulley system. For a single movable pulley, the constraint is typically a₁ = 2a₂.

Write Newton's second law for each mass. For m₁: T - m₁g = m₁a₁. For m₂: m₂g - T = m₂a₂. These equations describe the net force on each mass and their respective accelerations.

Substitute the acceleration constraint (a₁ = 2a₂) into the equations. Replace a₁ in the equation for m₁ with 2a₂ to express everything in terms of a single variable (a₂). This step ensures the system of equations is consistent with the physical setup.

Solve the system of equations to find an expression for the acceleration of m₁ (a₁). Combine the two equations by eliminating T, and then solve for a₂. Once a₂ is determined, use the relationship a₁ = 2a₂ to express the acceleration of m₁ in terms of m₁, m₂, and g.

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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

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 relationship is expressed mathematically as F = ma, where F is the net force, m is the mass, and a is the acceleration. Understanding this law is crucial for analyzing the forces acting on m₁ and determining its acceleration.

Acceleration Constraint

In systems involving pulleys and multiple masses, the acceleration constraint refers to the relationship between the accelerations of different masses connected by the pulley system. For example, if one mass moves up, another mass connected by the same rope must move down, leading to a specific ratio of their accelerations. Recognizing this constraint is essential for deriving the correct expression for the acceleration of m₁.

Massless and Frictionless Pulleys

Massless and frictionless pulleys simplify the analysis of mechanical systems by eliminating additional forces that would otherwise complicate calculations. A massless pulley does not contribute to the overall mass of the system, while a frictionless pulley ensures that there is no energy loss due to friction. This idealization allows for a straightforward application of Newton's laws and the acceleration constraint in solving for the acceleration of m₁.

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Textbook Question

FIGURE P7.47 shows a 200 g hamster sitting on an 800 g wedge-shaped block. The block, in turn, rests on a spring scale. An extra-fine lubricating oil having μ_s = μ_k = 0 is sprayed on the top surface of the block, causing the hamster to slide down. Friction between the block and the scale is large enough that the block does not slip on the scale. What does the scale read, in grams, as the hamster slides down?

Textbook Question

What is the acceleration of the 3.0 kg block in FIGURE CP7.55 across the frictionless table? Hint: Think carefully about the acceleration constraint.

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Textbook Question

The 1.0 kg physics book in FIGURE P7.40 is connected by a string to a 500 g coffee cup. The book is given a push up the slope and released with a speed of 3.0 m/s. The coefficients of friction are μ_s = 0.50 and μ_k = 0.20. At the highest point, does the book stick to the slope, or does it slide back down?

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Textbook Question

A house painter uses the chair-and-pulley arrangement of FIGURE P7.45 to lift himself up the side of a house. The painter's mass is 70 kg and the chair's mass is 10 kg. With what force must he pull down on the rope in order to accelerate upward at 0.20 m/s².

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