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Ch 06: Dynamics I: Motion Along a Line
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook
All textbooksKnight Calc 5th EditionCh 06: Dynamics I: Motion Along a LineProblem 40a
Chapter 6, Problem 40a

An accident victim with a broken leg is being placed in traction. The patient wears a special boot with a pulley attached to the sole. The foot and boot together have a mass of 4.0 kg, and the doctor has decided to hang a 6.0 kg mass from the rope. The boot is held suspended by the ropes, as shown in FIGURE P6.40, and does not touch the bed. Determine the amount of tension in the rope by using Newton's laws to analyze the hanging mass. Hint: If the pulleys are frictionless, which we will assume, the tension in the rope is constant from one end to the other.

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Step 1: Analyze the forces acting on the hanging mass. The mass is 6.0 kg, and the force due to gravity acting on it is given by F = m * g, where m is the mass and g is the acceleration due to gravity (approximately 9.8 m/s²). This force is the weight of the mass.
Step 2: Recognize that the tension in the rope is constant throughout because the pulleys are frictionless. This means the tension in the rope supporting the hanging mass is the same as the tension in the rope connected to the boot.
Step 3: Apply Newton's second law to the hanging mass. Since the mass is stationary, the net force acting on it is zero. Therefore, the upward tension in the rope must equal the downward gravitational force acting on the mass.
Step 4: Write the equation for equilibrium: T = m * g, where T is the tension in the rope, m is the mass of the hanging object (6.0 kg), and g is the acceleration due to gravity (9.8 m/s²). Substitute the values into the equation to find the tension.
Step 5: Interpret the result. The tension calculated in the previous step is the same throughout the rope, including the section holding the boot. This tension is responsible for keeping the boot suspended in equilibrium.

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Key Concepts

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

Newton's Laws of Motion

Newton's Laws of Motion describe the relationship between the motion of an object and the forces acting on it. The first law states that an object at rest will remain at rest unless acted upon by a net external force. The second law quantifies this relationship, stating that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass (F=ma). The third law states that for every action, there is an equal and opposite reaction.
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Tension in a Rope

Tension is the force transmitted through a rope or string when it is pulled tight by forces acting from opposite ends. In a frictionless pulley system, the tension is constant throughout the rope. This means that the tension acting on the hanging mass is equal to the weight of the mass minus any other forces acting on the system, such as the weight of the boot in this scenario.
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Equilibrium of Forces

An object is in equilibrium when the net force acting on it is zero, meaning all forces are balanced. In the context of the traction system, the forces acting on the boot and the hanging mass must balance out. This involves considering the gravitational force acting on the masses and the tension in the rope, allowing us to set up equations to solve for unknowns, such as the tension in the rope.
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