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Chapter 9: Problem 43

Wheel \(A\) has three times the moment of inertia about its axis of rotation as wheel \(B .\) Wheel \(B\) 's angular speed is four times that of wheel \(A\). (a) Which wheel has the greater rotational kinetic energy? (b) If \(K_{A}\) and \(K_{B}\) are the rotational kinetic energies of the wheels, what is \(K_{A} / K_{B} ?\)

Short Answer

Expert verified
Wheel \(B\) has greater rotational kinetic energy. The ratio \(K_{A} / K_{B} = \frac{3}{4}\).

Step by step solution

01

Read and Understand the Problem

In the problem, two wheels \(A\) and \(B\) are given. The moment of inertia of wheel \(A\) is three times that of wheel \(B\) denoted as \(I_{A} = 3I_{B}\). Wheel \(B\) 's angular speed is four times that of wheel \(A\) denoted as \(\omega_{B} = 4\omega_{A}\). We are asked to find which wheel has the greater rotational kinetic energy and the value of \(K_{A} / K_{B}\).
02

Formulate Equations for Kinetic Energy

Begin by using the equation of rotational kinetic energy \(K = \frac{1}{2} I \omega^2\). Substituting in the relations between moments of inertia and angular speeds between the two wheels, we get \(K_{A} = \frac{1}{2} 3I_{B} \omega_A^2\) and \(K_{B} = \frac{1}{2} I_{B} (4 \omega_{A})^2\). Simplify these to get \(K_{A} = \frac{3}{2} I_{B} \omega_{A}^2\) and \(K_{B} = 2 I_{B} \omega_{A}^2\).
03

Determine Which Wheel Has Greater Kinetic Energy

Just by comparing \(K_{A}\) and \(K_{B}\), it can be seen that \(K_{B}\) is larger than \(K_{A}\) because \(2 I_{B} \omega_{A}^2 > \frac{3}{2} I_{B} \omega_{A}^2\). Thus, wheel \(B\) has greater kinetic energy.
04

Compute the Ratio \(K_{A} / K_{B}\)

Calculate the ratio \(K_{A} / K_{B} = (\frac{3}{2} I_{B} \omega_{A}^2) / (2 I_{B} \omega_{A}^2) = \frac{3}{4}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Moment of Inertia
Moment of inertia, often represented by the symbol 'I', is a measure of an object's resistance to changes in its rotation rate. It's an inertial property of a rigid body that quantifies how the distribution of mass is spaced relative to an axis of rotation. Just like mass is a measure of how much an object resists linear acceleration, moment of inertia is that measure for rotational acceleration.

In the given exercise, wheel A has a moment of inertia which is three times that of wheel B, mathematically represented as \(I_{A} = 3I_{B}\). This fundamental concept dictates how much torque is required for a wheel to reach a certain angular speed, or simply put, how 'hard' it is to spin the wheel. The greater the moment of inertia, the more energy it will take to change the wheel's rotational speed.
Angular Speed
Angular speed, denoted by the symbol '\(\omega\)', refers to how fast an object rotates or revolves relative to another point, which is typically the object's center of rotation. It's analogous to linear speed but instead applies to rotational movement. Angular speed is measured in radians per second (rad/s).

In our exercise, wheel B's angular speed is four times that of wheel A (\(\omega_{B} = 4\omega_{A}\)). The equation \(K = \frac{1}{2} I \omega^2\) relates angular speed to rotational kinetic energy, demonstrating that the energy is not only dependent on the speed itself but also the square of it. This means that even a slight increase in angular speed can greatly increase the rotational kinetic energy.
Physics Problem Solving
Effective physics problem solving typically involves the clear understanding of concepts and formulas, identifying relevant information, and methodically applying logical steps to find the solution. In the context of this exercise, it primarily involved identifying the relationship between moment of inertia and angular speed as they contribute to rotational kinetic energy.

By breaking down the problem into smaller steps, one can systematically determine which wheel has greater kinetic energy and the exact ratio of their kinetic energies. Knowing the right equations to use and how to manipulate them plays a crucial role. The step-by-step solution provided demonstrates a logical progression from reading and understanding the problem, through to computing the final answer, which is a quintessential skill set in physics problem solving.

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Most popular questions from this chapter

An electric fan is turned off, and its angular velocity decreases uniformly from 500 rev \(/\) min to 200 rev \(/ \min\) in 4.00 s. (a) Find the angular acceleration in rev/s \(^{2}\) and the number of revolutions made by the motor in the 4.00 s interval. (b) How many more seconds are required for the fan to come to rest if the angular acceleration remains constant at the value calculated in part (a)?

A high-speed flywheel in a motor is spinning at 500 rpm when a power failure suddenly occurs. The flywheel has mass \(40.0 \mathrm{~kg}\) and diameter \(75.0 \mathrm{~cm}\). The power is off for \(30.0 \mathrm{~s}\), and during this time the flywheel slows due to friction in its axle bearings. During the time the power is off, the flywheel makes 200 complete revolutions. (a) At what rate is the flywheel spinning when the power comes back on? (b) How long after the beginning of the power failure would it have taken the flywheel to stop if the power had not come back on, and how many revolutions would the wheel have made during this time?

A wheel of diameter \(40.0 \mathrm{~cm}\) starts from rest and rotates with a constant angular acceleration of \(3.00 \mathrm{rad} / \mathrm{s}^{2}\). Compute the radial acceleration of a point on the rim for the instant the wheel completes its second revolution from the relationship (a) \(a_{\mathrm{rad}}=\omega^{2} r\) and (b) \(a_{\mathrm{rad}}=v^{2} / r\)

A thin, uniform rod is bent into a square of side length \(a\). If the total mass is \(M,\) find the moment of inertia about an axis through the center and perpendicular to the plane of the square. (Hint: Use the parallel-axis theorem.)

A uniform wheel in the shape of a solid disk is mounted on a frictionless axle at its center. The wheel has mass \(5.00 \mathrm{~kg}\) and radius \(0.800 \mathrm{~m} .\) A thin rope is wrapped around the wheel, and a block is suspended from the free end of the rope. The system is released from rest and the block moves downward. What is the mass of the block if the wheel turns through 8.00 revolutions in the first \(5.00 \mathrm{~s}\) after the block is released?
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