# 10.1 Rotational variables  (Page 6/11)

 Page 6 / 11

We now have a basic vocabulary for discussing fixed-axis rotational kinematics and relationships between rotational variables. We discuss more definitions and connections in the next section.

## Summary

• The angular position $\theta$ of a rotating body is the angle the body has rotated through in a fixed coordinate system, which serves as a frame of reference.
• The angular velocity of a rotating body about a fixed axis is defined as $\omega \left(\text{rad}\text{/}\text{s}\right)$ , the rotational rate of the body in radians per second. The instantaneous angular velocity of a rotating body $\omega =\underset{\text{Δ}t\to 0}{\text{lim}}\frac{\text{Δ}\omega }{\text{Δ}t}=\frac{d\theta }{dt}$ is the derivative with respect to time of the angular position $\theta$ , found by taking the limit $\text{Δ}t\to 0$ in the average angular velocity $\stackrel{–}{\omega }=\frac{\text{Δ}\theta }{\text{Δ}t}$ . The angular velocity relates ${v}_{\text{t}}$ to the tangential speed of a point on the rotating body through the relation ${v}_{\text{t}}=r\omega$ , where r is the radius to the point and ${v}_{\text{t}}$ is the tangential speed at the given point.
• The angular velocity $\stackrel{\to }{\omega }$ is found using the right-hand rule. If the fingers curl in the direction of rotation about a fixed axis, the thumb points in the direction of $\stackrel{\to }{\omega }$ (see [link] ).
• If the system’s angular velocity is not constant, then the system has an angular acceleration. The average angular acceleration over a given time interval is the change in angular velocity over this time interval, $\stackrel{–}{\alpha }=\frac{\text{Δ}\omega }{\text{Δ}t}$ . The instantaneous angular acceleration is the time derivative of angular velocity, $\alpha =\underset{\text{Δ}t\to 0}{\text{lim}}\frac{\text{Δ}\omega }{\text{Δ}t}=\frac{d\omega }{dt}$ . The angular acceleration $\stackrel{\to }{\alpha }$ is found by locating the angular velocity. If a rotation rate of a rotating body is decreasing, the angular acceleration is in the opposite direction to $\stackrel{\to }{\omega }$ . If the rotation rate is increasing, the angular acceleration is in the same direction as $\stackrel{\to }{\omega }$ .
• The tangential acceleration of a point at a radius from the axis of rotation is the angular acceleration times the radius to the point.

## Conceptual questions

A clock is mounted on the wall. As you look at it, what is the direction of the angular velocity vector of the second hand?

The second hand rotates clockwise, so by the right-hand rule, the angular velocity vector is into the wall.

What is the value of the angular acceleration of the second hand of the clock on the wall?

A baseball bat is swung. Do all points on the bat have the same angular velocity? The same tangential speed?

They have the same angular velocity. Points further out on the bat have greater tangential speeds.

The blades of a blender on a counter are rotating clockwise as you look into it from the top. If the blender is put to a greater speed what direction is the angular acceleration of the blades?

## Problems

Calculate the angular velocity of Earth.

A track star runs a 400-m race on a 400-m circular track in 45 s. What is his angular velocity assuming a constant speed?

$\omega =\frac{2\pi \phantom{\rule{0.2em}{0ex}}\text{rad}}{45.0\phantom{\rule{0.2em}{0ex}}\text{s}}=0.14\phantom{\rule{0.2em}{0ex}}\text{rad/s}$

A wheel rotates at a constant rate of $2.0\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{3}\phantom{\rule{0.2em}{0ex}}\text{rev}\text{/}\text{min}\phantom{\rule{0.2em}{0ex}}$ . (a) What is its angular velocity in radians per second? (b) Through what angle does it turn in 10 s? Express the solution in radians and degrees.

A particle moves 3.0 m along a circle of radius 1.5 m. (a) Through what angle does it rotate? (b) If the particle makes this trip in 1.0 s at a constant speed, what is its angular velocity? (c) What is its acceleration?

a. $\theta =\frac{s}{r}=\frac{3.0\phantom{\rule{0.2em}{0ex}}\text{m}}{1.5\phantom{\rule{0.2em}{0ex}}\text{m}}=2.0\phantom{\rule{0.2em}{0ex}}\text{rad}$ ; b. $\omega =\frac{2.0\phantom{\rule{0.2em}{0ex}}\text{rad}}{1.0\phantom{\rule{0.2em}{0ex}}\text{s}}=2.0\phantom{\rule{0.2em}{0ex}}\text{rad/s}$ ; c. $\frac{{v}^{2}}{r}=\frac{{\left(3.0\phantom{\rule{0.2em}{0ex}}\text{m/s}\right)}^{2}}{1.5\phantom{\rule{0.2em}{0ex}}\text{m}}=6.0\phantom{\rule{0.2em}{0ex}}\text{m}\text{/}{\text{s}}^{2}.$

A compact disc rotates at 500 rev/min. If the diameter of the disc is 120 mm, (a) what is the tangential speed of a point at the edge of the disc? (b) At a point halfway to the center of the disc?

Unreasonable results. The propeller of an aircraft is spinning at 10 rev/s when the pilot shuts off the engine. The propeller reduces its angular velocity at a constant $2.0\phantom{\rule{0.2em}{0ex}}\text{rad}\text{/}{\text{s}}^{2}$ for a time period of 40 s. What is the rotation rate of the propeller in 40 s? Is this a reasonable situation?

The propeller takes only $\text{Δ}t=\frac{\text{Δ}\omega }{\alpha }=\frac{0\phantom{\rule{0.2em}{0ex}}\text{rad/s}-10.0\left(2\pi \right)\phantom{\rule{0.2em}{0ex}}\text{rad/s}}{-2.0\phantom{\rule{0.2em}{0ex}}\text{rad}\text{/}{\text{s}}^{2}}=31.4\phantom{\rule{0.2em}{0ex}}\text{s}$ to come to rest, when the propeller is at 0 rad/s, it would start rotating in the opposite direction. This would be impossible due to the magnitude of forces involved in getting the propeller to stop and start rotating in the opposite direction.

A gyroscope slows from an initial rate of 32.0 rad/s at a rate of $0.700{\phantom{\rule{0.2em}{0ex}}\text{rad/s}}^{2}$ . How long does it take to come to rest?

On takeoff, the propellers on a UAV (unmanned aerial vehicle) increase their angular velocity from rest at a rate of $\omega =\left(25.0t\right)\phantom{\rule{0.2em}{0ex}}\text{rad}\text{/}\text{s}$ for 3.0 s. (a) What is the instantaneous angular velocity of the propellers at $t=2.0\phantom{\rule{0.2em}{0ex}}\text{s}$ ? (b) What is the angular acceleration?

a. $\omega =25.0\left(2.0\phantom{\rule{0.2em}{0ex}}\text{s}\right)=50.0\phantom{\rule{0.2em}{0ex}}\text{rad/s}$ ; b. $\alpha =\frac{d\omega }{dt}=25.0\phantom{\rule{0.2em}{0ex}}{\text{rad/s}}^{2}$

The angular position of a rod varies as $20.0{t}^{2}$ radians from time $t=0$ . The rod has two beads on it as shown in the following figure, one at 10 cm from the rotation axis and the other at 20 cm from the rotation axis. (a) What is the instantaneous angular velocity of the rod at $t=5\phantom{\rule{0.2em}{0ex}}\text{s}?$ (b) What is the angular acceleration of the rod? (c) What are the tangential speeds of the beads at $t=5\phantom{\rule{0.2em}{0ex}}\text{s}?$ (d) What are the tangential accelerations of the beads at $t=5\phantom{\rule{0.2em}{0ex}}\text{s}?$ (e) What are the centripetal accelerations of the beads at $t=5\phantom{\rule{0.2em}{0ex}}\text{s}?$

#### Questions & Answers

what is mass
ilance Reply
it is the quantity of matter in a substance....expressed in Kg as the SI unit.
Carloes
what is the universe of conductivity
Agbeyangi Reply
what is si unit of force?
Jose Reply
Kg m / s^2 You can see that this is the case from dimensional analysis of F = M a
Peter
Newton which is Kgm/s2
NAWED
newton(N)
Carloes
What's power in physics
Richard
newton(N)
Shahzab
Power is the rate of doing work. P=W/T
DOLLY
A car moving with a velocity of 54km/hr accelerates uniformly at the rate of 2mper seconds. Calculate the distance traveled from the place where acceleration began to where the velocity reaches 72km/hr and the time taken to cover d distance
Betty Reply
t=2.5s and s=43.75m
Ajayi
how to find amplitude of interfering wave
Roha Reply
how do we slove a pendulum clock shaft question
Seray Reply
An object of mass m moves with acceleration a down a rough incline.
Phyu Reply
If the average velocity of an object is zero in some time interval, what can you say about the displacement of the object fr that interval?
Phyu
what is force
Biggie Reply
you ever seen star wars?
Jake
Yes
okon
What is angular velocity
Amara Reply
refer to how fast an object rotates ot revolve to another point
samson
An object undergoes constant acceleration after starting from rest and then travels 5m in the first seconds .determine how far it will go in the next seconds
Sophy Reply
x=x0+v0t+1/2at^2
Grant
what is the deference between precision and accuracy
ShAmy
In measurement of a set, accuracy refers to closeness of the measurements to a specific value, while precision refers to the closeness of the measurements to each other.
Grant
Thank you Mr..
ShAmy
Are there is a difference between Error and Relative Error...
ShAmy
An aircraft flies 300km due east and 600km due north. determine the magnitude of its displacement
Sophy Reply
670.8km
iyiola
which formula did you use
Sophy
X^2=300^2+600^2
iyiola
what is Fermat principle.
Madhu Reply
find the angle of projection at which the horizontal range is twice the maximum height of a projectile
Akaji Reply
impulse by height fomula
Tigst Reply
Hello
Madue
what is impulse?
Tigst Reply
impulse is the integral of a force (F),over the interval for which it act
Agbeyangi

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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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