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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
All textbooksYoung & Freedman Calc 14th EditionCh 10: Dynamics of Rotational MotionProblem 10b
Chapter 10, Problem 10b

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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Identify the forces acting on the wheel: the tension in the cord (40.0 N) acts horizontally to the right, and the gravitational force (weight) acts vertically downward. The axle exerts a force on the wheel to balance these forces.
Calculate the gravitational force (weight) acting on the wheel using the formula: \( F_{gravity} = m \cdot g \), where \( m = 9.20 \text{ kg} \) and \( g = 9.81 \text{ m/s}^2 \).
Recognize that the wheel is in equilibrium, meaning the net force and net torque on the wheel are zero. Therefore, the force exerted by the axle must balance both the horizontal and vertical components of the forces acting on the wheel.
Determine the horizontal component of the force exerted by the axle. Since the only horizontal force is the tension in the cord, the horizontal component of the axle's force must be equal and opposite to this tension (40.0 N to the left).
Determine the vertical component of the force exerted by the axle. Since the only vertical force is the gravitational force, the vertical component of the axle's force must be equal and opposite to the gravitational force calculated in step 2.

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Key Concepts

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

Torque

Torque is the rotational equivalent of force, which causes an object to rotate around an axis. It is calculated as the product of the force applied and the distance from the axis of rotation (lever arm). In this problem, the torque is generated by the 40.0 N force applied tangentially to the wheel, causing it to rotate around its axle.
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Newton's Second Law for Rotation

Newton's Second Law for rotation states that the net torque acting on a body is equal to the product of its moment of inertia and its angular acceleration. For the wheel, the net torque due to the pulling force results in angular acceleration, which can be used to determine the force exerted by the axle by considering the wheel's moment of inertia.
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Equilibrium of Forces

In a system in equilibrium, the sum of forces and the sum of torques must be zero. For the wheel, the horizontal pull and the force exerted by the axle must balance each other to maintain equilibrium. Analyzing these forces helps determine the magnitude and direction of the force that the axle exerts on the wheel, considering both rotational and translational motion.
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Related Practice
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

Three forces are applied to a wheel of radius 0.350 m, as shown in Fig. E10.4. One force is perpendicular to the rim, one is tangent to it, and the other one makes a 40.0° angle with the radius. What is the net torque on the wheel due to these three forces for an axis perpendicular to the wheel and passing through its center?

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Textbook Question

The flywheel of an engine has moment of inertia 1.60 kg/m2 about its rotation axis. What constant torque is required to bring it up to an angular speed of 400 rev/min in 8.00 s, starting from rest?

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Textbook Question

(a) Calculate the magnitude of the angular momentum of the earth in a circular orbit around the sun. Is it reasonable to model it as a particle? Consult Appendix E and the astronomical data in Appendix F

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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 mass of the stone.

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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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