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Now let’s apply this problem-solving strategy to a few specific examples.

A spinning bicycle wheel

A bicycle mechanic mounts a bicycle on the repair stand and starts the rear wheel spinning from rest to a final angular velocity of 250 rpm in 5.00 s. (a) Calculate the average angular acceleration in rad/s 2 . (b) If she now hits the brakes, causing an angular acceleration of −87.3 rad/s 2 , how long does it take the wheel to stop?

Strategy

The average angular acceleration can be found directly from its definition α = Δ ω Δ t because the final angular velocity and time are given. We see that Δ ω = ω final ω initial = 250 rev/min and Δ t is 5.00 s. For part (b), we know the angular acceleration and the initial angular velocity. We can find the stopping time by using the definition of average angular acceleration and solving for Δ t , yielding

Δ t = Δ ω α .

Solution

  1. Entering known information into the definition of angular acceleration, we get
    α = Δ ω Δ t = 250 rpm 5.00 s .

    Because Δ ω is in revolutions per minute (rpm) and we want the standard units of rad/s 2 for angular acceleration, we need to convert from rpm to rad/s:
    Δ ω = 250 rev min · 2 π rad rev · 1 min 60 s = 26.2 rad s .

    Entering this quantity into the expression for α , we get
    α = Δ ω Δ t = 26.2 rad/s 5.00 s = 5.24 rad/s 2 .
  2. Here the angular velocity decreases from 26.2 rad/s (250 rpm) to zero, so that Δ ω is −26.2 rad/s, and α is given to be –87.3 rad/s 2 . Thus,
    Δ t = −26.2 rad/s −87.3 rad/s 2 = 0.300 s .

Significance

Note that the angular acceleration as the mechanic spins the wheel is small and positive; it takes 5 s to produce an appreciable angular velocity. When she hits the brake, the angular acceleration is large and negative. The angular velocity quickly goes to zero.

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Check Your Understanding The fan blades on a turbofan jet engine (shown below) accelerate from rest up to a rotation rate of 40.0 rev/s in 20 s. The increase in angular velocity of the fan is constant in time. (The GE90-110B1 turbofan engine mounted on a Boeing 777, as shown, is currently the largest turbofan engine in the world, capable of thrusts of 330–510 kN.)

(a) What is the average angular acceleration?

(b) What is the instantaneous angular acceleration at any time during the first 20 s?

Picture is a photo of an air turbine under the wing of an airplane.
(credit: “Bubinator”/ Wikimedia Commons)

a. 40.0 rev / s = 2 π ( 40.0 ) rad / s , α = Δ ω Δ t = 2 π ( 40.0 ) 0 rad/s 20.0 s = 2 π ( 2.0 ) = 4.0 π rad/ s 2 ; b. Since the angular velocity increases linearly, there has to be a constant acceleration throughout the indicated time. Therefore, the instantaneous angular acceleration at any time is the solution to 4.0 π rad/ s 2 .

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Wind turbine

A wind turbine ( [link] ) in a wind farm is being shut down for maintenance. It takes 30 s for the turbine to go from its operating angular velocity to a complete stop in which the angular velocity function is ω ( t ) = [ ( t s −1 −30.0 ) 2 / 100.0 ] rad / s . If the turbine is rotating counterclockwise looking into the page, (a) what are the directions of the angular velocity and acceleration vectors? (b) What is the average angular acceleration? (c) What is the instantaneous angular acceleration at t = 0.0 , 15.0 , 30.0 s ?

Figure is a drawing of a wind turbine that is rotating counterclockwise, as seen head on.
A wind turbine that is rotating counterclockwise, as seen head on.

Strategy

  1. We are given the rotational sense of the turbine, which is counterclockwise in the plane of the page. Using the right hand rule ( [link] ), we can establish the directions of the angular velocity and acceleration vectors.
  2. We calculate the initial and final angular velocities to get the average angular acceleration. We establish the sign of the angular acceleration from the results in (a).
  3. We are given the functional form of the angular velocity, so we can find the functional form of the angular acceleration function by taking its derivative with respect to time.

Solution

  1. Since the turbine is rotating counterclockwise, angular velocity ω points out of the page. But since the angular velocity is decreasing, the angular acceleration α points into the page, in the opposite sense to the angular velocity.
  2. The initial angular velocity of the turbine, setting t = 0 , is ω = 9.0 rad / s . The final angular velocity is zero, so the average angular acceleration is
    α = Δ ω Δ t = ω ω 0 t t 0 = 0 9.0 rad/s 30.0 0 s = −0.3 rad / s 2 .
  3. Taking the derivative of the angular velocity with respect to time gives α = d ω d t = ( t 30.0 ) / 50.0 rad / s 2
    α ( 0.0 s ) = −0.6 rad / s 2 , α ( 15.0 s ) = −0.3 rad / s 2 , and α ( 30.0 s ) = 0 rad/s .

Significance

We found from the calculations in (a) and (b) that the angular acceleration α and the average angular acceleration α are negative. The turbine has an angular acceleration in the opposite sense to its angular velocity.

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Questions & Answers

different types of wave
Yog Reply
longitudinal and transverse waves
Ravindra
a gun is kept in the state that it cannot move anywhere and the bullet is fired. Then what is the effect on the velocity of bullet and KE of gun ?
Gobinda Reply
want is meant by the term solar system
jafar Reply
it refers to the sun and all heavenly bodies revolving around it.
Danie
excatly...for sure
Arzoodan
in addition to Danie's, a solar system is a collection of planets and their moons, asteroids, and other objects bound together by the Star's gravitational force directly or indirectly.
Galiwango
what is meant by total internal reflection
Akshay Reply
what iw meant by total internal reflection
Akshay
Lorentz force?
jyotirmayee
study fibre optics. .you will get total internal reflection
Siddhansh
wha
jyotirmayee
what is Lorentz force?
jyotirmayee
a ray of light traveling at an angle of incidence greater than critical angle from denser to rarer medium is totally reflected back into the denser medium is called total internal reflaction
Manoj
motion in strat line where is this chapter
Vijaybhai Reply
what is Lorentz force?
jyotirmayee Reply
what is maxwell electromagnetic law?
jyotirmayee
at what angle should the two forces 2p and root 2p acts so that the resultant force is p root 10
Akshay Reply
what answer fir this
Akshay
what's the working difference between a dynamo and a pump?
Piyali Reply
a dynamo is basically a dc generator while pump is usually equipped with a motor
vedanth
a dynamo converts mechanical energy to electrical while a pump is opposite to that
vedanth
nice
Piyali
okay
Friday
why sea water looks bluish?
Piyali Reply
cuz the sky is blue...
Mehdi
you see the reflexion of the "blue" sky in the water
Mehdi
somewhere sea water turns green why?
Piyali
never seen bro... are u sure ?
Mehdi
ur answer was correct but due to the presence of phytoplankton color can be changed near the shore
Piyali
waaaww... you re awesome
Mehdi
because of the reflection of the sky
Friday
rays coming from the sun consist of all 7 colours ie.VIBGYOR. when the ray strikes surface of water,all colors gets absorbed by it except blue which gets reflected by it.so we find the sea water appearing bluish
Ritik
how can someone identify sea water from rain water
Oniyide
i need solutions of unuversity pbysucs volume 1
Vimla Reply
me too
Nirupam
Help us if anyone knows
Nirupam
bring questions
john
actually if u wanted whole book solution then u should buy the solution book
jyotirmayee
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Nirupam
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jyotirmayee
Kota Rajasthan
Nirupam
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jyotirmayee
for IIT
Nirupam
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jyotirmayee
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jyotirmayee
or ask questions here
jyotirmayee
thanks for your help
Nirupam
its fine,,most welcome
jyotirmayee
what is three dimensional coordinate system?
Sachindra Reply
considering the change of vectors in all three dimensions of space
vedanth
direction co-sign of vector questions
Vipin Reply
opposite
Sonu
opposite mean
Vipin
what is the different between action and reaction?
Asali Reply
action is external force. Reaction exists because of the external action. Reaction is mostly an internal force
Yoblaze
The difference is said in the word itself. There is no existance of reaction without an application of action. So action occur first then reaction. Reaction may produce in body itself or to other body i.e., it may be internal or external.
Amalesh
what is total energy
Israel Reply
kenetic energy + potential energy + rotational energy
Tony
any body or object can have it ..even in the molecular level
Tony
As you eat your way through a bag of chocolate chip cookies, you observe that each cookie is a circular disk of diameter 8.50 ± 0.02 cm and thickness 0.050 ± 0.005 cm. Find the average volume of a cookie and the uncertainty in the volume.
Mike Reply
what is the Greek name for calcium
Oniyide Reply
Practice Key Terms 5

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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