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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?

Strategy

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"} {}

Discussion

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

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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.)

Strategy

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.

Solution

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"} {}

Discussion

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 .

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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.

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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} } " " \( "constant "α, a \) } {}

Problems&Exercises

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

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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.)

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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

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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)
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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

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Questions & Answers

velocity is produce in fan...?
bilal Reply
how many electrons are there in 5 microcouloumb
Obed Reply
can a given total amount of mechanical energy be totally converted into heat energy..if so give example
Muhammad Reply
human running
Emmanuel
what is the fumula for calculating specific heat capacity, fusion,fission and vaporization?
Dohn Reply
Q=cm(∆t)
Emmanuel
Q=cm∆T
Muhammad
what is difference b/w vaporization and evaporation
Muhammad
evaporation is the process of extracting moisture while vaporization is process of becoming a vapor or gas
Emmanuel
From a molecular standpoint they are both cooling processes. Also, you may want to explore states of matter😊 #myTwoCents ~Shi~
Shii
cooling is a similarlity in both process I am confused in difference
Muhammad
1- Evaporation is a process where a liquid change to gas without reaching its boiling point. 2- Vaporization is a process where a liquid change to gas after reaching its boiling point. 3- Sublimation is a process where a solid changes into vapour without passing through a liquid state
Victor
I see. Evaporation is a type of vaporization, that occurs on the surface of a liquid as it changes into the gaseous phase before reaching its boiling point. hope that aids
Shii
vaporisation is cooling process while vaporization is heating process
Emmanuel
I mean to write evaporation is an heating process while vaporization is cooling process
Emmanuel
Yea here are two applications. 1- your wet washed clothes dry under the sun, the water EVAPORATES 2- when u are cooking, it reaches a point where u need to add more water because the water you added previously is getting dried. this is VAPORIZATION. Am not sure which is a cooling or heating process
Victor
vaporization occur only when the evaporation get to level where the above cloud is been (saturated) so cooling take place and started to change to liquid (eg rain fall)
Emmanuel
They are both properties of the same process so they're both cooling
Shii
what about sublimation? cooling or heating process?
Victor
exact
Muhammad
evaporation is the increase in kinetic energy of the liquid which can be gone by adding heat
Emmanuel
so its an heating process
Emmanuel
sublimation is when a solid change to gas
Emmanuel
evaporation is very definitely a cooling process. respectfully@Emmanuel when liquid turns to gas it requires more energy from its surroundings, this energy is in the form of heat, and when heat energy leaves the evaporating liquid it leaves it cooler. Thus, cooling process.
Shii
.
Shii
evaporation is very definitely a cooling process. respectfully@Emmanuel
Shii
kk
Emmanuel
You're right @Shi. I get your point
Victor
eascape velocity on the surface of Earth is 11.2 kms-1 the escape velocity on the surface of another planet of same mass as that of Earth but of 1/4 times of radius of Earth is a5.6kms-1 b11.2 kms-1 c22.4kms-1 d5.6ms-1
Muhammad
Emm.. is that a question? or..
Victor
it is McQ
Muhammad
a)5.6km/s
Alvis
c= Q/cm◇T
A.d
it's answer is 22.4
Muhammad
units...
Shii
vital
Shii
the time period of the artificial satellite is given by ?
raza
Why is there no 2nd harmonic in the classical electron orbit?
Shree Reply
how to reform magnet after been demagneted
Inuwa Reply
A petrol engine has a output of 20 kilowatts and uses 4.5 kg of fuel for each hour of running. The energy given out when 1 kg of petrol is burnt is 4.8 × 10 to the power of 7 Joules. a) What is the energy output of the engine every hour? b) What is the energy input of the engine every hour?
Morris Reply
Issac Newton devised a genius way to calculate changing quantities...
Shii
what is the error during taking work done of a body..
Aliyu Reply
what kind of error do you think? and work is held by which force?
Daniela
I am now in this group
smart
theory,laws,principles and what-a-view are not defined. why? you
Douglas Reply
A simple pendulum is used in a physics laboratory experiment to obtain an experimental value for the gravitational acceleration, g . A student measures the length of the pendulum to be 0.510 meters, displaces it 10 o from the equilibrium position, and releases it. Using a s
Emmanuel Reply
so what question are you passing across... sir?
Olalekan
Two masses of 2 kg and 4 kg are held with a compressed spring between them. If the masses are released, the spring will push them away from each other. If the smaller mass moves off with a velocity of 6m/s, what is the stored energy in the spring when it is compressed?
Emmanuel Reply
54 joule
babar
how?
rakesh
Reduce that two body problem into one body problem. Apply potential and k. E formula to get total energy of the system
rakesh
i dont think dere is any potential energy... by d virtue of no height present
Olalekan
there is compressed energy,dats only potential energy na?
rakesh
yes.. but... how will u approach that question without The Height in the question?
Olalekan
Can you explain how you get 54J?
Emmanuel
Because mine is 36J
Emmanuel
got 36J too
Douglas
OK the answer is 54J Babar is correct
Emmanuel
Conservation of Momentum
Emmanuel
woow i see.. can you give the formula for this
joshua
Two masses of 2 kg and 4 kg are held with a compressed spring between them. If the masses are released, the spring will push them away from each other. If the smaller mass moves off with a velocity of 6m/s, what is the stored energy in the spring when it is compressed? Asume there is no external force.
Emmanuel Reply
Please help!
Emmanuel
please help find dy/dx 2x-y/x+y
Inuwa
By using the Quotient Rule dy/dx = 3y/(x +y)²
Emmanuel
3y/(x+y)²
Emmanuel
may be by using MC^2=MC^2 and Total energy=kinetic energy +potential energy so 1st find kinetic energy and den find potential energy which is stored energy
rakesh
i think i m correct
rakesh
But how?
Emmanuel
3y/(x+y)²
Douglas
what's the big bang?
kwame Reply
yes what is it?
LamaBbake
it is the explanation of how the universe began
Zainab
yes
Ana
explain
Chinagorom
in
Chinagorom
it is a theory on how the universe began. to understand more I would suggest researching the topic online.
david
thanks guys
kwame
the Big Bang is actually ironic name given by a skeptic( at the time) common misconception is that there was some explosion when actually it's the phenomenon of the start of our continuous expansion of the universe. Brilliant
Shii
we all know E=MC^2 in this equation the mass can be converted in to energy and energy converted in to mass then i have studied the laws of energy and mass that these quantities are consvered then how it can be possible that mass converted energy ?. plz reply me with ans
A.d
hello
Gold
yes
bilal
ya but energy can't b created nor destroyed so where does it come from
Mohit
velocity is avaliable in fan..?
bilal
if a force of 12N is applied to load of 200g what us the work done
Joshua Reply
We can seek accelation first
Nancy
we are given f=12 m=200g which is 0.2kg now from 2nd law of newton a= f/m=60m/s*2 work done=force applied x displacement cos (theta) w= 12x60 =720nm/s*2
Mudang
this very interesting question very complicated for me, í need urgent help. 1,two buses A and B travel along the same road in the same direction from Harper city (asume They both started from the same point) to Monrovia. if bus A maintains a Speedy of 60km/h and bus B a Speedy of 75km/h, how many
mohammed
hours Will it take bus B to overtake bus A assuming bus B starts One hour after bus A started. what is the distance travelled by the buses when They meet?.
mohammed
pls í need help
mohammed
4000 work is done
Ana
speed=distance /time distance=speed/time
Ana
now use this formula
Ana
what's the answer then
Julius
great Mudang
Kossi
please Ana explain 4000 ?
babar
hey mudang there is a product of force and acceleration not force and displacement
babar
@Mohammed answer is 0.8hours or 48mins
Douglas
nice
A.d
its not possible
Olalekan
í want the working procedure
mohammed
the answer is given but how Will One arrive at it. the answers are 4hours and 300m.
mohammed
physics is the science that studies the non living nature
isidor Reply
ancient greek language physis = nature
isidor
Practice Key Terms 1

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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