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Ch. 04 - Dynamics: Newton's Laws of Motion
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 04 - Dynamics: Newton's Laws of MotionProblem 63a
Chapter 4, Problem 63a

The double Atwood machine shown in Fig. 4–55 has frictionless, massless pulleys and cords. Determine the acceleration of masses mA, mB, and mC.
Diagram of a double Atwood machine with labeled masses mA, mB, mC and forces FT_A, FT_C.

Verified step by step guidance
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Identify the forces acting on each mass. For mass m_A, the forces are its weight (m_A * g) and the tension in the cord (T1). For mass m_B, the forces are its weight (m_B * g) and the tensions in the cords (T1 and T2). For mass m_C, the forces are its weight (m_C * g) and the tension in the cord (T2).
Write Newton's second law for each mass. For m_A: m_A * a_A = T1 - m_A * g. For m_B: m_B * a_B = T1 - T2 - m_B * g. For m_C: m_C * a_C = T2 - m_C * g. Here, a_A, a_B, and a_C are the accelerations of the respective masses.
Relate the accelerations of the masses using the constraints of the system. Since the pulleys and cords are massless and frictionless, the movement of one mass affects the others. For example, if m_A moves up, m_B and m_C must move in a way that conserves the length of the cords. This gives a relationship like a_A = 2 * a_B = 2 * a_C (depending on the pulley configuration).
Solve the system of equations obtained from Newton's second law and the acceleration relationships. Substitute the expressions for T1 and T2 from one equation into the others to eliminate the tensions and solve for the accelerations a_A, a_B, and a_C in terms of the masses m_A, m_B, and m_C and the gravitational acceleration g.
Simplify the final expressions for the accelerations. The results will be in terms of m_A, m_B, m_C, and g, showing how the masses and gravity determine the motion of the system.

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 principle is fundamental in analyzing systems like the double Atwood machine, where multiple masses are connected by pulleys. By applying this law, one can derive the equations of motion for each mass and determine their accelerations.
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Intro to Forces & Newton's Second Law

Tension in Cords

In a system involving pulleys and cords, tension is the force transmitted through the cord when it is pulled tight by forces acting at either end. In the double Atwood machine, the tension in the cords affects the acceleration of the masses. Understanding how tension varies in different segments of the cord is crucial for setting up the equations needed to solve for the accelerations of the masses.
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Free Body Diagrams

Free body diagrams (FBDs) are graphical representations used to visualize the forces acting on an object. In the context of the double Atwood machine, drawing FBDs for each mass helps identify all the forces, including gravitational force and tension. This visual aid is essential for applying Newton's laws and solving for unknown quantities like acceleration.
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Related Practice
Textbook Question

An object is hanging by a string from your rearview mirror. While you are accelerating at a constant rate from rest to 28 m/s in 5.0 s, what angle θ does the string make with the vertical? See Fig. 4–46.

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Textbook Question

Consider the system shown in Fig. 4–68 with mA = 8.2kg and mB = 11.5kg. The angles θA = 59° and θB = 32°. In the absence of friction, what force F→\(\overrightarrow{F}\) would be required to pull the masses at a constant velocity up the fixed inclines?

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Textbook Question

The double Atwood machine shown in Fig. 4–55 has frictionless, massless pulleys and cords. Determine the tensions FTA and FTC in the cords.

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Textbook Question

Determine a formula for the acceleration of the system shown in Fig. 4–49 (see Problem 55) if the cord has a non-negligible mass mC. Specify in terms of ℓA and ℓB , the lengths of cord from the respective masses to the pulley. (The total cord length is ℓA + ℓB.)

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Textbook Question

Three mountain climbers who are roped together in a line are ascending an icefield inclined at 29° to the horizontal (Fig. 4–67). The last climber slips, pulling the second climber off his feet. The first climber is able to hold them both. If each climber has a mass of 75 kg, calculate the tension in each of the two sections of rope between the three climbers. Ignore friction between the ice and the fallen climbers.

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Textbook Question

As shown in Fig. 4–48, five balls (masses 2.00, 2.05, 2.10, 2.15, 2.20 kg) hang from a crossbar. Each mass is supported by '5-lb test' fishing line which will break when its tension force exceeds 22.2 N (5.00lb). When this device is placed in an elevator, which accelerates upward, only the lines attached to the 2.05 and 2.00 kg masses do not break. Within what range is the elevator's acceleration?

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