Continuing to Payment will take you to apayment page

Purchasing
SparkNotes PLUS
for a group?

Get Annual Plans at a discount when you buy 2 or more!

Price

$24.99$18.74/subscription + tax

Subtotal $37.48 + tax

Save 25%
on 2-49 accounts

Save 30%
on 50-99 accounts

Want 100 or more?
Contact us
for a customized plan.

Continuing to Payment will take you to apayment page

Your Plan

Payment Details

Payment Details

Payment Summary

SparkNotes Plus

You'll be billed after your free trial ends.

7-Day Free Trial

Not Applicable

Renews June 3, 2024May 27, 2024

Discounts (applied to next billing)

DUE NOW

US $0.00

SNPLUSROCKS20 | 20%Discount

This is not a valid promo code.

Discount Code(one code per order)

SparkNotes PLUS
Annual Plan - Group Discount

Qty: 00

SubtotalUS $0,000.00

Discount (00% off)
-US $000.00

TaxUS $XX.XX

DUE NOWUS $1,049.58

SparkNotes Plus subscription is $4.99/month or $24.99/year as selected above. The free trial period is the first 7 days of your subscription. TO CANCEL YOUR SUBSCRIPTION AND AVOID BEING CHARGED, YOU MUST CANCEL BEFORE THE END OF THE FREE TRIAL PERIOD. You may cancel your subscription on your Subscription and Billing page or contact Customer Support at custserv@bn.com. Your subscription will continue automatically once the free trial period is over. Free trial is available to new customers only.

Choose Your Plan

Your Free Trial Starts Now!

For the next 7 days, you'll have access to awesome PLUS stuff like AP English test prep, No Fear Shakespeare translations and audio, a note-taking tool, personalized dashboard, & much more!

Thank You!

You’ve successfully purchased a group discount. Your group members can use the joining link below to redeem their group membership. You'll also receive an email with the link.

No URL

Copy

Members will be prompted to log in or create an account to redeem their group membership.

Thanks for creating a SparkNotes account! Continue to start your free trial.

We're sorry, we could not create your account. SparkNotes PLUS is not available in your country. See what countries we’re in.

There was an error creating your account. Please check your payment details and try again.

Please wait while we process your payment

Your PLUS subscription has expired

We’d love to have you back! Renew your subscription to regain access to all of our exclusive, ad-free study tools.

Renew your subscription to regain access to all of our exclusive, ad-free study tools.

Having established the dynamics of rotational motion, we can now extend our
study to work and energy. Given what we already know, the equations governing
energetics are quite easy to derive. Finally, with the equations that we have
derived, we will be able to describe the complicated situations involving
combined rotational and translational motion.

Work

Given our definition of work as W = Fs, can we generate an expression for work
done on a rotational system? To derive our expression we begin by taking the
simplest case: when the force applied to a particle in rotational motion is
perpendicular to the radius of the particle. In this orientation, the force
applied is parallel to the displacement of the particle, and would exert the
maximum work. Given this situation the work done is simply W = Fs, where s is
the arc length that the force acts through in a given period of time. Recall,
however, that arc length can also be expressed in terms of the angle swept out
by the arc: s = rμ. Our expression for work in this simple case becomes:

W = Frθ = τμ

Since Fr gives us our torque, we can simplify our expression in terms of only
τ
and μ.

What if the force is not perpendicular to the radius of the particle? Let the
angle between the force vector and the radius vector be θ, as shown
below.

To compute the work we calculate the component of the force acting in the
direction of the particle's displacement. In this case, this quantity is simply
F sinθ. Again, this force acts over an arc length given by rμ.
Thus the work is given by:

W = (F sinθ)(rμ) = (Fr sinθ)μ

Recall that

τ = Fr sinθ

Thus W = τμ
Surprisingly enough, this equation is exactly the same as our special case when
the force acted perpendicular to the radius! In any case, the work done by a
given force is equal to the torque it exerts multiplied by the angular
displacement.

For you calculus types, there is also an equation for work done by variable
torques. Instead of deriving it, we can just state it, as it is quite similar
to the equation in the linear case:

W = τdμ

Thus we have quickly gone through deriving our expression for work. The next
thing after work we studied in linear motion was kinetic energy, and it is to
this topic that we turn.

Rotational Kinetic Energy

Consider a wheel spinning in place. Clearly the wheel is moving, and has a
kinetic energy attached to it. But the wheel is not engaged in translational
motion. How do we calculate the kinetic energy of the wheel? Our answer is
similar to how we calculated the result of a net torque on a body: by summing
over each particle.