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Chapter 11: Problem 33

Rotational Inertia of Wheel Calculate the rotational inertia of a wheel that has a kinetic energy of \(24400 \mathrm{~J}\) when rotating at 602 rev \(/ \mathrm{min}\).

Short Answer

Expert verified
Rotational inertia of the wheel is approximately 12.315 kg m^2.

Step by step solution

01

- Convert rotational speed to radians per second

First, convert the rotational speed from revolutions per minute (rev/min) to radians per second (rad/s). Use the conversion factor 1 rev = 2蟺 radians and 1 minute = 60 seconds. Thus, the speed in rad/s is: \[\text{angular speed} = 602 \text{ rev/min} \times \frac{2\text{蟺 radians}}{1 \text{ rev}} \times \frac{1 \text{ min}}{60 \text{ s}}\] Calculate the value.
02

- Calculate the angular speed

Substitute the values: \[ \text{angular speed} = 602 \times \frac{2\text{蟺}}{60} \text{ rad/s}\] Simplify the expression: \[ \text{angular speed} \therefore 62.969 \text{ rad/s}\]
03

- Relate to kinetic energy formula

The kinetic energy of a rotating object is given by the formula: \[ K.E. = \frac{1}{2} I \text{蠅}^2 \] where \( I \) is the rotational inertia and \( \text{蠅} \) is the angular speed.
04

- Solve for rotational inertia

Rearrange the formula to solve for \( I \): \[ I = \frac{2 \times \text{K.E.}}{\text{蠅}^2} \] Substitute \(\text{ K.E. } = 24400 \text{ J} \) and \(\text{ 蠅 } = 62.969 \text{ rad/s} \): \[ I = \frac{2 \times 24400}{(62.969)^2} \] Calculate the value.
05

- Calculate the rotational inertia

Substitute and simplify the expression: \[ I = \frac{2 \times 24400}{3962.378} \] \[ I \therefore 12.315 \text{ kg m}^2 \]

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Key Concepts

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

Angular Speed Conversion
When you encounter a rotational problem, it's common to need to convert angular speed from one unit to another. In this case, we begin with revolutions per minute (rev/min) and convert it to radians per second (rad/s).
Recall, 1 revolution = 2蟺 radians, and 1 minute = 60 seconds. These conversion factors help us transform the given speed.
For our problem, we start with 602 rev/min. We multiply by \(\frac{2蟺 \text{ radians}}{1 \text{ rev}}\) to convert revolutions to radians, and then by \(\frac{1 \text{ min}}{60 \text{ s}}\) to convert minutes to seconds.
So, the calculation goes: \[ \text{angular speed} = 602 \times \frac{2蟺}{60} \text{ rad/s} \] . Doing the math, we obtain approximately 62.969 rad/s.
This converted speed will be used in further calculations.
Kinetic Energy Formula
The kinetic energy of a rotating object is given by a formula similar to linear kinetic energy but adapted for rotation.
The formula is: \[ K.E. = \frac{1}{2} I \text{蠅}^2 \] where \( K.E. \) is kinetic energy, \( I \) is rotational inertia, and \( 蠅 \) is angular speed.
This formula tells us that the kinetic energy depends on how much inertia the object has as well as how fast it's spinning.
Understanding this formula is key, especially when you're given kinetic energy and need to find rotational inertia. Rearranging to solve for \( I \), we get: \[ I = \frac{2 \times \text{K.E.}}{\text{蠅}^2} \].
Here, we'll substitute \( K.E. = 24400 \text{ J} \) and \( 蠅 = 62.969 \text{ rad/s} \) into the formula, performing these calculations will let us solve for rotational inertia.
Rotational Dynamics
Rotational dynamics involves the study of objects in rotation and how various forces and torques affect their motion.
It鈥檚 analogous to linear dynamics but with rotational quantities like angular speed, torque, and rotational inertia.
One fundamental concept in rotational dynamics is rotational inertia (or moment of inertia), denoted by \( I \).
This tells us how hard it is to change the rotational motion of an object. Mathematically, it depends on how an object's mass is distributed relative to the axis of rotation.
Combining rotational inertia with angular speed gives us deeper insight into the rotational kinetic energy of objects.
In the given exercise, we've seen how to use the converted angular speed and known kinetic energy to find rotational inertia.
Understanding the principles of rotational dynamics allows you to tackle various problems involving rotating systems, making this concept indispensable for mastering physics of rotation.

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Most popular questions from this chapter

Hands of a Clock What is the rotational speed of (a) the second hand, (b) the minute hand, and (c) the hour hand of a smoothly running analog watch? Answer in radians per second.

A Flywheel Has a Rotational Velocity At \(t_{1}=0\), a flywheel has a rotational velocity of \(4.7 \mathrm{rad} / \mathrm{s}\), a rotational acceleration of \(-0.25\) \(\mathrm{rad} / \mathrm{s}^{2}\), and a reference line at \(\theta_{1}=0 .\) (a) Through what maximum angle \(\theta^{\max }\) will the reference line turn in the positive direction? For what length of time will the reference line turn in the positive direction? At what times will the reference line be at (b) \(\theta=\frac{1}{2} \theta^{\max }\) and (c) \(\theta=-10.5\) rad (consider both positive and negative values of \(t\) )? (d) Graph \(\theta\) versus \(t\), and indicate the answers to \((\mathrm{a}),(\mathrm{b})\), and \((\mathrm{c})\) on the graph.

Communications \(\quad\) Satellite \(A\) communications satellite is a solid cylinder with mass \(1210 \mathrm{~kg}\), diameter \(1.21 \mathrm{~m}\), and length \(1.75 \mathrm{~m}\). Prior to launching from the shuttle cargo bay, it is set spinning at \(1.52 \mathrm{rev} / \mathrm{s}\) about the cylinder axis (Fig. \(11-27)\). Calculate the satellite's (a) rotational inertia about the rotation axis and (b) rotational kinetic energy.

Heavy Flywheel A heavy flywheel rotating on its central axis is slowing down because of friction in its bearings. At the end of the first minute of slowing, its rotational speed is \(0.90\) of its initial rotational speed of 250 rev/min. Assuming a constant rotational acceleration, find its rotational speed at the end of the second minute.

A Diver A diver makes \(2.5\) revolutions on the way from a \(10-\mathrm{m}\) high platform to the water. Assuming zero initial vertical velocity, find the diver's average rotational velocity during a dive.
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