# 10.2 Kinematics of rotational motion  (Page 3/4)

 Page 3 / 4

## 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 $0\text{.}\text{250}\phantom{\rule{0.25em}{0ex}}{\text{rad/s}}^{2}$ . 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 $x$ , and in (b) we are asked to find $\omega$ and $v$ . We are given the number of revolutions $\theta$ , the radius of the wheels $r$ , and the angular acceleration $\alpha$ .

Solution for (a)

The distance $x$ is very easily found from the relationship between distance and rotation angle:

$\theta =\frac{x}{r}.$

Solving this equation for $x$ yields

$x=\mathrm{r\theta .}$

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:

$\theta =\left(\text{200}\phantom{\rule{0.25em}{0ex}}\text{rev}\right)\frac{2\pi \phantom{\rule{0.25em}{0ex}}\text{rad}}{\text{1 rev}}=\text{1257}\phantom{\rule{0.25em}{0ex}}\text{rad}.$

Now we can substitute the known values into $x=\mathrm{r\theta }$ to find the distance the train moved down the track:

$x=\mathrm{r\theta }=\left(0.350 m\right)\left(\text{1257 rad}\right)=\text{440}\phantom{\rule{0.25em}{0ex}}\text{m}.$

Solution for (b)

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

${\omega }^{2}={{\omega }_{0}}^{2}+2\text{αθ}$

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

We can find the linear velocity of the train, $v$ , through its relationship to $\omega$ :

$v=\mathrm{r\omega }=\left(0.350 m\right)\left(\text{25.1 rad/s}\right)=\text{8.77 m/s}.$

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.

## 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 $\theta$ , and then the linear distance $x$ traveled. $\theta =\overline{\omega }t$ can be used to find $\theta$ because $\stackrel{-}{\omega }$ is given to be 6.0 rpm.

Solution

Entering known values into $\theta =\overline{\omega }t$ gives

$\theta =\stackrel{-}{\omega }t=\left(\text{6.0 rpm}\right)\left(\text{2.0 min}\right)=\text{12 rev}.$

As always, it is necessary to convert revolutions to radians before calculating a linear quantity like $x$ from an angular quantity like $\theta$ :

$\theta =\left(\text{12 rev}\right)\left(\frac{2\pi \phantom{\rule{0.25em}{0ex}}\text{rad}}{\text{1 rev}}\right)=\text{75}\text{.4 rad.}$

Now, using the relationship between $x$ and $\theta$ , we can determine the distance traveled:

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 ( ${x}_{0}$ and ${t}_{0}$ are initial values), and the average angular velocity $\stackrel{-}{\omega }$ and average velocity $\stackrel{-}{v}$ are defined as follows:

## 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) $80\phantom{\rule{0.25em}{0ex}}{\text{rad/s}}^{2}$

(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 .

(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 .

(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 ?

(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?

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 , 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) $6{\text{00 rad/s}}^{2}$

c) 21.0 m/s

How submarines floats one water the same time sink in water
A submarine has the ability to float and sink. The ability to control buoyancy comes from the submarine'strim or ballast tanks which can be filled with either water or air, depending on whether the submarine needs to floator sink. When the submarine floats it means its trim tanks are filled with air
Arif
what is work
Force times distance
Karanja
product of force and distance...
Arif
Is physics a natural science?
what is the difference between a jet engine and a rocket engine.
explain the relationship between momentum and force
A moment is equivalent multiplied by the length passing through the point of reaction and that is perpendicular to the force
Karanja
How to find Squirrel frontal area from it's surface area?
how do we arrange the electronic configuration of elements
hi guys i am an elementary student
hi
Dancan
hello
are you an elementary student too?
benedict
no bro
yes
Che
hi
Miranwa
yes
Miranwa
welcome
Miranwa
what is the four equation of motion
Miranwa
what is strain?
SAMUEL
Change in dimension per unit dimension is called strain. Ex - Change in length per unit length l/L.
ABHIJIT
strain is the ratio of extension to length..=e/l...it has no unit because both are in meters and they cancel each other
How is it possible for one to drink a cold drink from a straw?
most possible as it is for you to drink your wine from your straw
Selina
state the law of conservation of energy
energy can neither be destroy or created,but can be change from one form to another
dare
yeah
Toheeb
it can neither be created nor destroyed
Toheeb
its so sample question dude
Muhsin
what is the difference between a principle and a law?
where are from you wendy .?
ghulam
philippines
Mary
why?
Mary
you are beautiful
ghulam
are you physics student
ghulam
laws are ment to be broken
Ge
hehe ghulam where r u from?
Muhsin
yes
dare
principle are meant to be followed
dare
south Africa
dare
here Nigeria
Toheeb
principle is a rule or law of nature, or the basic idea on how the laws of nature are applied.
Ayoka
Rules are meant to be broken while principals to be followed
Karanja
principle is a rule or law of nature, or the basic idea on how the laws of nature are applied.
tathir
what is momentum?
is the mass times velocity of an object
True
it is the product of mass and velocity of an object.
The momentum possessed by a body is generally defined as the product of its mass and velocity m×v
Usman
momentum is the product of the mass of a body of its velocity
Ugbesia
what about kg it is changing or not
no mass is the quantity or amount of body so it remains constant everywhere
Ahsan
yes
Siyanbola
remains constant
taha
mass of an object is always constant. and that is universally applied.
Shii
mass of a body never changes but the weight can change due to variance of gravity at different points of the world
Saheed
what is hookes law
Joshua
mass of an object does not change
SAMUEL
Is weight a scalar quantity
weight is actually a force of gravity with which earth attracts us downwards so it is a vector quantity. and it has both direction and magnitude
Ahsan
ty
Denise
weight is the earth pull of the body
Ugbesia
why does weight change but not mass?
Theo
Theo, the mass of an object can change but it depends on how you define that object. First, you need to know that mass is the amount of matter an object has, and weight is mass*gravity (the "force" that attracts object A to the object B mass).
Nicolas
So if you face object A with object B, you will get a different result than facing object A with object C, so the weight of object A changes but not its mass.
Nicolas
Now, if you have an object and you take a part away from it, you are changing it mass. Lets use the human body and fat loss process as an example.
Nicolas
When you lose weight by doing exercise, you are being attracted by the same object before and after losing weight so the change of weight is related to a change of mass not a change of gravity.
Nicolas
The explanation of this is simple, we are composed of smaller particles, which are itself objects, so the loose of mass of an object actually is the separation of one object is two different ones.
Nicolas
But if you define an object because of its form and characteristics and not the amount of mass, then the object is the same but you have taken a part of it mass away.
Nicolas
Theo, weight =mass. gravity, here mass is fixed everywhere but gravity change in different places so weight change not mass.
ABHIJIT
yup weight changes and mass does not. That's why we're 1/3 our weight on the moon
clifford
weight is the product of mass × velocity w=m×v = m(v-u) but v=u+1/2at^ weight is a scalar quantity mass of an obj is the amount of particles that obj cont
Usman
mass is fixed always while weight is dynamic
Usman
Why does water wet glass but mercury does not?
Yusuf
thanks guys
Theo
Yusuf Shuaibu, for water the Adhessive force between water molecules and glass is greater than the cohessive force between it's own molecules but for Mercury the cohessive force will be greater in comparison with adhessive force. For this water wet glass but Mercury does not.
ABHIJIT
in electrostatic e bonite rod electron is static. they cannot flow to other. because static. is it correct?
Is weight a scalar quantity
esther
wieght is the vector
ghulam
yes
Mohet
Yes
Karanja