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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 88c
Chapter 4, Problem 88c

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 F→\(\overrightarrow{F}\), what is the tension in the string?
Diagram of two masses on inclined planes with angles 59° and 32°, illustrating tension in a connecting string.

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Identify the forces acting on each mass. For m_A, the forces are its weight (m_A * g), the tension in the string (T), and the component of the normal force along the incline. For m_B, the forces are its weight (m_B * g) and the tension in the string (T).
Break the forces into components along the incline for m_A and along the vertical for m_B. For m_A, the component of its weight along the incline is m_A * g * sin(θ_A). For m_B, the weight acts directly downward, so no trigonometric decomposition is needed.
Write the equations of motion for the system. Since the system is in equilibrium (no external force F→ and no acceleration), the net force along the direction of motion for both masses must be zero. This gives: T = m_A * g * sin(θ_A) for m_A and T = m_B * g - m_B * g * sin(θ_B) for m_B.
Set the two expressions for tension (T) equal to each other, as the string is the same for both masses. This gives: m_A * g * sin(θ_A) = m_B * g - m_B * g * sin(θ_B).
Solve for T (tension in the string) by isolating T in the equation. Simplify the equation to express T in terms of the given masses, angles, and gravitational acceleration (g).

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Key Concepts

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

Tension in a String

Tension is the force transmitted through a string, rope, or cable when it is pulled tight by forces acting from opposite ends. In a system involving pulleys or inclined planes, tension can vary depending on the masses involved and the angles of inclination. Understanding how to calculate tension is crucial for analyzing forces in static or dynamic systems.
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Energy & Power of Waves on Strings

Forces on Inclined Planes

When objects are placed on inclined planes, the gravitational force acting on them can be resolved into two components: one parallel to the incline and one perpendicular to it. The angle of inclination affects these components, influencing the net force acting on the object. This concept is essential for determining the forces at play in the system described in the question.
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Intro to Inclined Planes

Equilibrium Conditions

For a system to be in equilibrium, the sum of all forces acting on it must equal zero. This means that the forces in both the horizontal and vertical directions must balance out. In the context of the question, applying the conditions of equilibrium allows us to solve for unknowns, such as the tension in the string, by setting up equations based on the forces acting on the masses.
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Related Practice
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

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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