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 31a

Chapter 10, Problem 31a

A 2.80-kg grinding wheel is in the form of a solid cylinder of radius 0.100 m. What constant torque will bring it from rest to an angular speed of 1200 rev/min in 2.5 s?

Verified step by step guidance

First, convert the angular speed from revolutions per minute (rev/min) to radians per second (rad/s). Use the conversion factor: 1 rev = 2π rad and 1 min = 60 s.

Calculate the angular acceleration (α) using the formula: α = (ω_f - ω_i) / t, where ω_f is the final angular speed, ω_i is the initial angular speed (0 rad/s since it starts from rest), and t is the time (2.5 s).

Determine the moment of inertia (I) for the solid cylinder using the formula: I = (1/2) * m * r^2, where m is the mass (2.80 kg) and r is the radius (0.100 m).

Use the relationship between torque (τ), moment of inertia (I), and angular acceleration (α): τ = I * α. Substitute the values of I and α to find the torque.

Ensure all units are consistent and check the calculations for any errors. The torque calculated will be the constant torque required to bring the wheel to the desired angular speed in the given time.

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Key Concepts

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

Torque

Torque is a measure of the force that can cause an object to rotate about an axis. It is calculated as the product of the force and the distance from the axis of rotation, known as the lever arm. In this problem, torque is needed to change the angular speed of the grinding wheel from rest to a specified speed.

Moment of Inertia

Moment of inertia is a property of a body that defines its resistance to angular acceleration, depending on the mass distribution relative to the axis of rotation. For a solid cylinder, 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 achieve the desired angular acceleration.

Angular Kinematics

Angular kinematics involves the study of rotational motion parameters such as angular velocity and angular acceleration. The problem requires calculating the angular acceleration using the change in angular speed and time, which is then used to find the torque. The formula α = (ω_final - ω_initial) / t helps in determining the angular acceleration needed to solve the problem.

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