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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).

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 .

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} } " " \( "constant "α, a \) } {}

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

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?

Questions & Answers

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Source:  OpenStax, Unit 8 - rotational motion. OpenStax CNX. Feb 22, 2016 Download for free at https://legacy.cnx.org/content/col11970/1.1
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