Textbook Question
A 2.0-m-long, 500 g rope pulls a 10 kg block of ice across a horizontal, frictionless surface. The block accelerates at 2.0 m/s2. How much force pulls forward on he rope? Assume that the rope is perfectly horizontal.
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6. Intro to Forces (Dynamics)
Forces in Connected Systems of Objects
Problem 63a
Textbook Question
The double Atwood machine shown in Fig. 4–55 has frictionless, massless pulleys and cords. Determine the acceleration of masses mA, mB, and mC.
Verified step by step guidance1
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.
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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 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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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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