Ch 10: Dynamics of Rotational Motion

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 10: Dynamics of Rotational Motion

Problem 12a

Chapter 10, Problem 12a

A stone is suspended from the free end of a wire that is wrapped around the outer rim of a pulley, similar to what is shown in Fig. 10.10. The pulley is a uniform disk with mass 10.0 kg and radius 30.0 cm and turns on frictionless bearings. You measure that the stone travels 12.6 m in the first 3.00 s starting from rest. Find the mass of the stone.

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First, identify the given values: the mass of the pulley (10.0 kg), the radius of the pulley (30.0 cm or 0.3 m), the distance traveled by the stone (12.6 m), and the time taken (3.00 s). The stone starts from rest.

Calculate the linear acceleration of the stone using the kinematic equation: \( s = ut + \frac{1}{2}at^2 \), where \( s \) is the distance traveled, \( u \) is the initial velocity (0 m/s), \( a \) is the acceleration, and \( t \) is the time. Rearrange to solve for \( a \).

Relate the linear acceleration of the stone to the angular acceleration of the pulley using the formula: \( a = r\alpha \), where \( r \) is the radius of the pulley and \( \alpha \) is the angular acceleration. Solve for \( \alpha \).

Apply Newton's second law for rotation to the pulley: \( \tau = I\alpha \), where \( \tau \) is the torque, \( I \) is the moment of inertia of the pulley, and \( \alpha \) is the angular acceleration. For a uniform disk, \( I = \frac{1}{2}mr^2 \). Calculate \( I \) and use it to find \( \tau \).

Use the relationship between torque and force: \( \tau = Fr \), where \( F \) is the force exerted by the stone. Since \( F = mg \) (where \( m \) is the mass of the stone and \( g \) is the acceleration due to gravity), solve for \( m \) using the previously calculated torque.

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Key Concepts

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

Rotational Motion

Rotational motion involves objects that rotate around an axis. In this scenario, the pulley acts as a uniform disk, and understanding its rotational dynamics is crucial. The angular acceleration of the pulley is related to the linear acceleration of the stone, which can be calculated using kinematic equations.

Newton's Second Law

Newton's Second Law states that the force acting on an object is equal to the mass of the object multiplied by its acceleration (F = ma). This principle helps determine the tension in the wire and the gravitational force acting on the stone, which are essential for calculating the stone's mass.

Conservation of Energy

The conservation of energy principle states that energy cannot be created or destroyed, only transformed. In this problem, the potential energy of the stone is converted into kinetic energy of both the stone and the rotating pulley. Understanding this energy transformation helps in analyzing the system's dynamics and solving for the stone's mass.

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Related Practice

Textbook Question

A cord is wrapped around the rim of a solid uniform wheel 0.250 m in radius and of mass 9.20 kg. A steady horizontal pull of 40.0 N to the right is exerted on the cord, pulling it off tangentially from the wheel. The wheel is mounted on frictionless bearings on a horizontal axle through its center. Find the magnitude and direction of the force that the axle exerts on the wheel.

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Textbook Question

A 2.00-kg textbook rests on a frictionless, horizontal surface. A cord attached to the book passes over a pulley whose diameter is 0.150 m, to a hanging book with mass 3.00 kg. The system is released from rest, and the books are observed to move 1.20 m in 0.800 s. What is the tension in each part of the cord?

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Textbook Question

A machine part has the shape of a solid uniform sphere of mass 225 g and diameter 3.00 cm. It is spinning about a frictionless axle through its center, but at one point on its equator it is scraping against metal, resulting in a friction force of 0.0200 N at that point. How long will it take to decrease its rotational speed by 22.5 rad/s?

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Textbook Question

A stone is suspended from the free end of a wire that is wrapped around the outer rim of a pulley, similar to what is shown in Fig. 10.10. The pulley is a uniform disk with mass 10.0 kg and radius 30.0 cm and turns on frictionless bearings. You measure that the stone travels 12.6 m in the first 3.00 s starting from rest. Find the tension in the wire.

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Textbook Question

A 15.0-kg bucket of water is suspended by a very light rope wrapped around a solid uniform cylinder 0.300 m in diameter with mass 12.0 kg. The cylinder pivots on a frictionless axle through its center. The bucket is released from rest at the top of a well and falls 10.0 m to the water. With what speed does the bucket strike the water?

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Textbook Question

A machine part has the shape of a solid uniform sphere of mass 225 g and diameter 3.00 cm. It is spinning about a frictionless axle through its center, but at one point on its equator, it is scraping against metal, resulting in a friction force of 0.0200 N at that point. Find its angular acceleration.

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