Ch 12: Rotation of a Rigid Body

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 12: Rotation of a Rigid Body

Problem 65

Chapter 12, Problem 65

The 2.0 kg, 30-cm-diameter disk in FIGURE P12.65 is spinning at 300 rpm. How much friction force must the brake apply to the rim to bring the disk to a halt in 3.0 s?

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Determine the initial angular velocity of the disk in radians per second. Convert the given angular velocity from revolutions per minute (300 rpm) to radians per second using the formula: \( \omega = \frac{2\pi \times \text{rpm}}{60} \).

Calculate the angular deceleration required to bring the disk to a halt in 3.0 seconds. Use the kinematic equation for angular motion: \( \alpha = \frac{\Delta \omega}{\Delta t} \), where \( \Delta \omega \) is the change in angular velocity and \( \Delta t \) is the time interval.

Determine the moment of inertia of the disk. For a solid disk, the moment of inertia is given by \( I = \frac{1}{2} m r^2 \), where \( m \) is the mass of the disk and \( r \) is its radius (half the diameter).

Calculate the torque required to produce the angular deceleration. Use the relationship \( \tau = I \alpha \), where \( \tau \) is the torque, \( I \) is the moment of inertia, and \( \alpha \) is the angular deceleration.

Relate the torque to the friction force applied at the rim. The torque is given by \( \tau = F_r r \), where \( F_r \) is the friction force and \( r \) is the radius of the disk. Solve for \( F_r \) to find the required friction force.

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Key Concepts

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

Angular Velocity

Angular velocity is a measure of how quickly an object rotates around an axis, typically expressed in radians per second. In this problem, the disk's initial angular velocity can be calculated from its given speed of 300 revolutions per minute (rpm), which must be converted to radians per second to analyze the motion effectively.

Moment of Inertia

The moment of inertia is a property of a body that quantifies its resistance to angular acceleration about an axis. For a solid disk, it is calculated using the formula I = 0.5 * m * r^2, where m is the mass and r is the radius. This concept is crucial for determining how much torque is needed to stop the disk.

Torque and Friction Force

Torque is the rotational equivalent of linear force and is calculated as the product of the force applied and the distance from the axis of rotation. In this scenario, the friction force applied at the rim of the disk generates torque that opposes the disk's rotation, and understanding this relationship is essential to calculate the required friction force to bring the disk to a halt within the specified time.

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Textbook Question

Blocks of mass m₁ and m₂ are connected by a massless string that passes over the pulley in FIGURE P12.64. The pulley turns on frictionless bearings. Mass m₁ slides on a horizontal, frictionless surface. Mass m₂ is released while the blocks are at rest. Assume the pulley is massless. Find the acceleration of m₁ and the tension in the string. This is a Chapter 7 review problem.

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Textbook Question

Blocks of mass m₁ and m₂ are connected by a massless string that passes over the pulley in FIGURE P12.64. The pulley turns on frictionless bearings. Mass m₁ slides on a horizontal, frictionless surface. Mass m₂ is released while the blocks are at rest. Suppose the pulley has mass m_p and radius R. Find the acceleration of m₁ and the tensions in the upper and lower portions of the string. Verify that your answers agree with part a if you set m_p = 0.

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Textbook Question

A 30-cm-diameter, 1.2 kg solid turntable rotates on a 1.2-cm-diameter, 450 g shaft at a constant 33 rpm. When you hit the stop switch, a brake pad presses against the shaft and brings the turntable to a halt in 15 seconds. How much friction force does the brake pad apply to the shaft?

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Textbook Question

Your engineering team has been assigned the task of measuring the properties of a new jet-engine turbine. You've previously determined that the turbine's moment of inertia is 2.6 kg m². The next job is to measure the frictional torque of the bearings. Your plan is to run the turbine up to a predetermined rotation speed, cut the power, and time how long it takes the turbine to reduce its rotation speed by 50%. Your data are given in the table. Draw an appropriate graph of the data and, from the slope of the best-fit line, determine the frictional torque.

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Textbook Question

FIGURE P12.63 shows a 15 kg cylinder held at rest on a 20° slope. What is the magnitude of the static friction force?

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Textbook Question

Flywheels are large, massive wheels used to store energy. They can be spun up slowly, then the wheel's energy can be released quickly to accomplish a task that demands high power. An industrial flywheel has a 1.5 m diameter and a mass of 250 kg. Its maximum angular velocity is 1200 rpm. How much energy is stored in the flywheel?

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