Log out Go to App
Go to App

Learning Materials

Features

Discover

Chapter 8: Problem 1

Verify that \(\frac{1}{2} I \omega^{2}\) has dimensions of energy.

Short Answer

Expert verified
Answer: The dimensions of the formula \(\frac{1}{2} I \omega^{2}\) are (M × L² × T⁻²), which represent dimensions of energy.

Step by step solution

01

Identify dimensions of each variable

In the given formula \(\frac{1}{2} I \omega^{2}\), we have two variables, I and ω. I represents the moment of inertia, which has dimensions of mass (M) times distance squared (L²). ω represents angular velocity, which has dimensions of inverse time (T⁻¹).
02

Determine the dimensions of the formula

Now we need to determine the dimensions of the formula itself. In order to do this, we should determine the dimensions of \(\omega^{2}\) first: The dimensions of \(\omega^{2}\) would be (T⁻¹)² = T⁻². Now, we can find the dimensions of the entire formula by multiplying the dimensions of I and \(\omega^{2}\): (M × L²) × (T⁻²).
03

Compare dimensions with dimensions of energy

Finally, we need to compare the dimensions of our formula with the dimensions of energy. The dimensions of energy are mass (M) times distance squared (L²) divided by time squared (T²), or (M × L² × T⁻²). Comparing the dimensions of our formula and energy, we see that they are the same: (M × L² × T⁻²). Thus, we can conclude that the formula \(\frac{1}{2} I \omega^{2}\) has dimensions of energy.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

An experimental flywheel, used to store energy and replace an automobile engine, is a solid disk of mass \(200.0 \mathrm{kg}\) and radius $0.40 \mathrm{m} .\( (a) What is its rotational inertia? (b) When driving at \)22.4 \mathrm{m} / \mathrm{s}(50 \mathrm{mph}),$ the fully energized flywheel is rotating at an angular speed of \(3160 \mathrm{rad} / \mathrm{s} .\) What is the initial rotational kinetic energy of the flywheel? (c) If the total mass of the car is \(1000.0 \mathrm{kg},\) find the ratio of the initial rotational kinetic energy of the flywheel to the translational kinetic energy of the car. (d) If the force of air resistance on the car is \(670.0 \mathrm{N},\) how far can the car travel at a speed of $22.4 \mathrm{m} / \mathrm{s}(50 \mathrm{mph})$ with the initial stored energy? Ignore losses of mechanical energy due to means other than air resistance.
A figure skater is spinning at a rate of 1.0 rev/s with her arms outstretched. She then draws her arms in to her chest, reducing her rotational inertia to \(67 \%\) of its original value. What is her new rate of rotation?
A 2.0 -kg uniform flat disk is thrown into the air with a linear speed of \(10.0 \mathrm{m} / \mathrm{s} .\) As it travels, the disk spins at 3.0 rev/s. If the radius of the disk is \(10.0 \mathrm{cm},\) what is the magnitude of its angular momentum?
A chain pulls tangentially on a \(40.6-\mathrm{kg}\) uniform cylindrical gear with a tension of 72.5 N. The chain is attached along the outside radius of the gear at \(0.650 \mathrm{m}\) from the axis of rotation. Starting from rest, the gear takes 1.70 s to reach its rotational speed of 1.35 rev/s. What is the total frictional torque opposing the rotation of the gear?
A house painter is standing on a uniform, horizontal platform that is held in equilibrium by two cables attached to supports on the roof. The painter has a mass of \(75 \mathrm{kg}\) and the mass of the platform is \(20.0 \mathrm{kg} .\) The distance from the left end of the platform to where the painter is standing is \(d=2.0 \mathrm{m}\) and the total length of the platform is $5.0 \mathrm{m}$ (a) How large is the force exerted by the left-hand cable on the platform? (b) How large is the force exerted by the right-hand cable?
See all solutions

Recommended explanations on Physics Textbooks

Turning Points in Physics

Read Explanation

Wave Optics

Read Explanation

Measurements

Read Explanation

Engineering Physics

Read Explanation

Thermodynamics

Read Explanation

Rotational Dynamics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free