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Calculating the slow acceleration of trains and their wheels

Large freight trains accelerate very slowly. Suppose one such train accelerates from rest, giving its 0.350-m-radius wheels an angular acceleration of 0 . 250 rad/s 2 size 12{0 "." "250"`"rad/s" rSup { size 8{2} } } {} . After the wheels have made 200 revolutions (assume no slippage): (a) How far has the train moved down the track? (b) What are the final angular velocity of the wheels and the linear velocity of the train?


In part (a), we are asked to find x size 12{x} {} , and in (b) we are asked to find ω size 12{ω} {} and v size 12{v} {} . We are given the number of revolutions θ size 12{θ} {} , the radius of the wheels r size 12{r} {} , and the angular acceleration α size 12{α} {} .

Solution for (a)

The distance x size 12{x} {} is very easily found from the relationship between distance and rotation angle:

θ = x r . size 12{θ= { {x} over {r} } } {}

Solving this equation for x size 12{x} {} yields

x = rθ. size 12{x=rθ.} {}

Before using this equation, we must convert the number of revolutions into radians, because we are dealing with a relationship between linear and rotational quantities:

θ = 200 rev rad 1 rev = 1257 rad . size 12{θ= left ("200"" rev" right ) { {2π" rad"} over {"1 rev"} } ="1257"" rad"} {}

Now we can substitute the known values into x = size 12{x=rθ} {} to find the distance the train moved down the track:

x = = 0.350 m 1257 rad = 440 m . size 12{x=rθ= left (0 "." "350"`m right ) left ("1257"" rad" right )="440"" m"} {}

Solution for (b)

We cannot use any equation that incorporates t to find ω , because the equation would have at least two unknown values. The equation ω 2 = ω 0 2 + 2 αθ will work, because we know the values for all variables except ω :

ω 2 = ω 0 2 + 2 αθ

Taking the square root of this equation and entering the known values gives

ω = 0 + 2 ( 0 . 250  rad/s 2 ) ( 1257  rad ) 1 / 2 = 25.1 rad/s. alignl { stack { size 12{ω= left [0+2 \( 0 "." "250"" rad/s" rSup { size 8{2} } \) \( "1257"" rad" \) right ]rSup { size 8{1/2} } "." } {} # ="25" "." 1" rad/s" {}} } {}

We can find the linear velocity of the train, v size 12{v} {} , through its relationship to ω size 12{ω} {} :

v = = 0.350 m 25.1 rad/s = 8.77 m/s . size 12{v=rω= left (0 "." "350"" m" right ) left ("25" "." 1" rad/s" right )=8 "." "77"" m/s"} {}


The distance traveled is fairly large and the final velocity is fairly slow (just under 32 km/h).

There is translational motion even for something spinning in place, as the following example illustrates. [link] shows a fly on the edge of a rotating microwave oven plate. The example below calculates the total distance it travels.

The figure shows a fly that has landed on the rotating plate of the microwave. The direction of rotation of the plate, omega, is counterclockwise and is shown with an arrow.
The image shows a microwave plate. The fly makes revolutions while the food is heated (along with the fly).

Calculating the distance traveled by a fly on the edge of a microwave oven plate

A person decides to use a microwave oven to reheat some lunch. In the process, a fly accidentally flies into the microwave and lands on the outer edge of the rotating plate and remains there. If the plate has a radius of 0.15 m and rotates at 6.0 rpm, calculate the total distance traveled by the fly during a 2.0-min cooking period. (Ignore the start-up and slow-down times.)


First, find the total number of revolutions θ size 12{θ} {} , and then the linear distance x size 12{x} {} traveled. θ = ω ¯ t size 12{θ= {overline {ωt}} } {} can be used to find θ size 12{θ} {} because ω - size 12{ { bar {ω}}} {} is given to be 6.0 rpm.


Entering known values into θ = ω ¯ t size 12{θ= {overline {ωt}} } {} gives

θ = ω - t = 6.0 rpm 2.0 min = 12 rev .

As always, it is necessary to convert revolutions to radians before calculating a linear quantity like x size 12{x} {} from an angular quantity like θ size 12{θ} {} :

θ = 12 rev 2 π rad 1 rev = 75 .4 rad. size 12{θ= left ("12"" rev" right ) left ( { {2π" rad"} over {"1 rev"} } right )="75" "." 4" rad"} {}

Now, using the relationship between x size 12{x} {} and θ size 12{θ} {} , we can determine the distance traveled:

x = = 0 . 15  m 75 . 4  rad = 11  m . size 12{x=rθ= left (0 "." "15"" m" right ) left ("75" "." 4" rad" right )="11" "." 3" m"} {}


Quite a trip (if it survives)! Note that this distance is the total distance traveled by the fly. Displacement is actually zero for complete revolutions because they bring the fly back to its original position. The distinction between total distance traveled and displacement was first noted in One-Dimensional Kinematics .

Rotational kinematics has many useful relationships, often expressed in equation form. Are these relationships laws of physics or are they simply descriptive? (Hint: the same question applies to linear kinematics.)

Rotational kinematics (just like linear kinematics) is descriptive and does not represent laws of nature. With kinematics, we can describe many things to great precision but kinematics does not consider causes. For example, a large angular acceleration describes a very rapid change in angular velocity without any consideration of its cause.

Section summary

  • Kinematics is the description of motion.
  • The kinematics of rotational motion describes the relationships among rotation angle, angular velocity, angular acceleration, and time.
  • Starting with the four kinematic equations we developed in the One-Dimensional Kinematics , we can derive the four rotational kinematic equations (presented together with their translational counterparts) seen in [link] .
  • In these equations, the subscript 0 denotes initial values ( x 0 size 12{x rSub { size 8{0} } } {} and t 0 size 12{t rSub { size 8{0} } } {} are initial values), and the average angular velocity ω - size 12{ { bar {ω}}} {} and average velocity v - size 12{ { bar {v}}} {} are defined as follows:
    ω ¯ = ω 0 + ω 2  and  v ¯ = v 0 + v 2 . size 12{ {overline {ω}} = { {ω rSub { size 8{0} } +ω} over {2} } " and " {overline {v}} = { {v rSub { size 8{0} } +v} over {2} } " " \( "constant "α, a \) } {}


With the aid of a string, a gyroscope is accelerated from rest to 32 rad/s in 0.40 s.

(a) What is its angular acceleration in rad/s 2 ?

(b) How many revolutions does it go through in the process?

(a) 80 rad/s 2 size 12{80 rad/s" rSup { size 8{2} } } {}

(b) 1.0 rev

Suppose a piece of dust finds itself on a CD. If the spin rate of the CD is 500 rpm, and the piece of dust is 4.3 cm from the center, what is the total distance traveled by the dust in 3 minutes? (Ignore accelerations due to getting the CD rotating.)

A gyroscope slows from an initial rate of 32.0 rad/s at a rate of 0 . 700  rad/s 2 size 12{0 "." "700"`"rad/s" rSup { size 8{2} } } {} .

(a) How long does it take to come to rest?

(b) How many revolutions does it make before stopping?

(a) 45.7 s

(b) 116 rev

During a very quick stop, a car decelerates at 7 . 00  m/s 2 size 12{7 "." "00"`"m/s" rSup { size 8{2} } } {} .

(a) What is the angular acceleration of its 0.280-m-radius tires, assuming they do not slip on the pavement?

(b) How many revolutions do the tires make before coming to rest, given their initial angular velocity is 95 . 0  rad/s size 12{"95" "." 0`"rad/s"} {} ?

(c) How long does the car take to stop completely?

(d) What distance does the car travel in this time?

(e) What was the car’s initial velocity?

(f) Do the values obtained seem reasonable, considering that this stop happens very quickly?

The figure shows the left arm of a man with tattoo imprints and wearing a glove. He is circulating a yo-yo toy, which is in mid air and connected by the string to his hand. Some people are standing in the background watching the yo-yo trick.
Yo-yos are amusing toys that display significant physics and are engineered to enhance performance based on physical laws. (credit: Beyond Neon, Flickr)

Everyday application: Suppose a yo-yo has a center shaft that has a 0.250 cm radius and that its string is being pulled.

(a) If the string is stationary and the yo-yo accelerates away from it at a rate of 1 . 50  m/s 2 size 12{1 "." "50"`"m/s" rSup { size 8{2} } } {} , what is the angular acceleration of the yo-yo?

(b) What is the angular velocity after 0.750 s if it starts from rest?

(c) The outside radius of the yo-yo is 3.50 cm. What is the tangential acceleration of a point on its edge?

a) 6 00 rad/s 2 size 12{ {underline {6"00 rad/s" rSup { size 8{2} } }} } {}

b) 450 rad/s

c) 21.0 m/s

Questions & Answers

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
what is Nano technology ?
Bob Reply
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The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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Damian Reply
what king of growth are you checking .?
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Stoney Reply
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Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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biomolecules are e building blocks of every organics and inorganic materials.
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Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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Damian Reply
absolutely yes
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
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Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
Mapo Reply
The formulation of new memories is sometimes called ________, and the process of bringing up old memories is called ________.
Mapo Reply
How we can toraidal magnetic field
Aditya Reply
How we can create polaidal magnetic field
Mykayuh Reply
Because I'm writing a report and I would like to be really precise for the references
Gre Reply
where did you find the research and the first image (ECG and Blood pressure synchronized)? Thank you!!
Gre Reply
Practice Key Terms 1

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Source:  OpenStax, Physics 101. OpenStax CNX. Jan 07, 2013 Download for free at http://legacy.cnx.org/content/col11479/1.1
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