<< Chapter < Page Chapter >> Page >

We give a strategy for using this equation when analyzing rotational motion.

Problem-solving strategy: work-energy theorem for rotational motion

  1. Identify the forces on the body and draw a free-body diagram. Calculate the torque for each force.
  2. Calculate the work done during the body’s rotation by every torque.
  3. Apply the work-energy theorem by equating the net work done on the body to the change in rotational kinetic energy.

Let’s look at two examples and use the work-energy theorem to analyze rotational motion.

Rotational work and energy

A 12.0 N · m torque is applied to a flywheel that rotates about a fixed axis and has a moment of inertia of 30.0 kg · m 2 . If the flywheel is initially at rest, what is its angular velocity after it has turned through eight revolutions?

Strategy

We apply the work-energy theorem. We know from the problem description what the torque is and the angular displacement of the flywheel. Then we can solve for the final angular velocity.

Solution

The flywheel turns through eight revolutions, which is 16 π radians. The work done by the torque, which is constant and therefore can come outside the integral in [link] , is

W A B = τ ( θ B θ A ) .

We apply the work-energy theorem:

W A B = τ ( θ B θ A ) = 1 2 I ω B 2 1 2 I ω A 2 .

With τ = 12.0 N · m , θ B θ A = 16.0 π rad , I = 30.0 kg · m 2 , and ω A = 0 , we have

12.0 N-m ( 16.0 π rad ) = 1 2 ( 30.0 kg · m 2 ) ( ω B 2 ) 0 .

Therefore,

ω B = 6.3 rad / s .

This is the angular velocity of the flywheel after eight revolutions.

Significance

The work-energy theorem provides an efficient way to analyze rotational motion, connecting torque with rotational kinetic energy.

Got questions? Get instant answers now!

Rotational work: a pulley

A string wrapped around the pulley in [link] is pulled with a constant downward force F of magnitude 50 N. The radius R and moment of inertia I of the pulley are 0.10 m and 2.5 × 10 −3 kg-m 2 , respectively. If the string does not slip, what is the angular velocity of the pulley after 1.0 m of string has unwound? Assume the pulley starts from rest.

Figure A shows a string wrapped around a pulley of radius R. The pulley is pulled down with a force F. Figure B shows free body that is pulled down with forces F and Mg and is pushed up with force B .
(a) A string is wrapped around a pulley of radius R . (b) The free-body diagram.

Strategy

Looking at the free-body diagram, we see that neither B , the force on the bearings of the pulley, nor M g , the weight of the pulley, exerts a torque around the rotational axis, and therefore does no work on the pulley. As the pulley rotates through an angle θ , F acts through a distance d such that d = R θ .

Solution

Since the torque due to F has magnitude τ = R F , we have

W = τ θ = ( F R ) θ = F d .

If the force on the string acts through a distance of 1.0 m, we have, from the work-energy theorem,

W A B = K B K A F d = 1 2 I ω 2 0 ( 50.0 N ) ( 1.0 m ) = 1 2 ( 2.5 × 10 −3 kg-m 2 ) ω 2 .

Solving for ω , we obtain

ω = 200.0 rad / s .
Got questions? Get instant answers now!

Power for rotational motion

Power always comes up in the discussion of applications in engineering and physics. Power for rotational motion is equally as important as power in linear motion and can be derived in a similar way as in linear motion when the force is a constant. The linear power when the force is a constant is P = F · v . If the net torque is constant over the angular displacement, [link] simplifies and the net torque can be taken out of the integral. In the following discussion, we assume the net torque is constant. We can apply the definition of power derived in Power to rotational motion. From Work and Kinetic Energy , the instantaneous power (or just power) is defined as the rate of doing work,

Questions & Answers

I'm interested in biological psychology and cognitive psychology
Tanya Reply
what does preconceived mean
sammie Reply
physiological Psychology
Nwosu Reply
How can I develope my cognitive domain
Amanyire Reply
why is communication effective
Dakolo Reply
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
Wekolamo Reply
please i need answer
Wekolamo
because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).
Manix Reply
Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!
ARC
A child is a member of community not society elucidate ?
JESSY Reply
Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?
Simon Reply
compare and contrast skinner's perspective on personality development on freud
namakula Reply
Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning
war
explain how nature and nurture affect the development and later the productivity of an individual.
Amesalu Reply
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
Zyryn Reply
good👍
Jonathan
and having a good philosophy of the world is like a sandwich and a peanut butter 👍
Jonathan
generally amnesi how long yrs memory loss
Kelu Reply
interpersonal relationships
Abdulfatai Reply
What would be the best educational aid(s) for gifted kids/savants?
Heidi Reply
treat them normal, if they want help then give them. that will make everyone happy
Saurabh
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 2

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'University physics volume 1' conversation and receive update notifications?

Ask