# 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 $t=5\phantom{\rule{0.2em}{0ex}}\text{s}?$

definition of inertia
the reluctance of a body to start moving when it is at rest and to stop moving when it is in motion
charles
An inherent property by virtue of which the body remains in its pure state or initial state
Kushal
why current is not a vector quantity , whereas it have magnitude as well as direction.
why
daniel
the flow of current is not current
fitzgerald
bcoz it doesn't satisfy the algabric laws of vectors
Shiekh
The Electric current can be defined as the dot product of the current density and the differential cross-sectional area vector : ... So the electric current is a scalar quantity . Scalars are related to tensors by the fact that a scalar is a tensor of order or rank zero .
Kushal
what is binomial theorem
what is binary operations
Tollum
What is the formula to calculat parallel forces that acts in opposite direction?
position, velocity and acceleration of vector
hi
peter
hi
daniel
hi
Vedisha
*a plane flies with a velocity of 1000km/hr in a direction North60degree east.find it effective velocity in the easterly and northerly direction.*
imam
hello
Lydia
hello Lydia.
Sackson
What is momentum
isijola
hello
A rail way truck of mass 2400kg is hung onto a stationary trunk on a level track and collides with it at 4.7m|s. After collision the two trunk move together with a common speed of 1.2m|s. Calculate the mass of the stationary trunk
I need the solving for this question
philip
is the eye the same like the camera
I can't understand
Suraia
Josh
I think the question is that ,,, the working principal of eye and camera same or not?
Sardar
yes i think is same as the camera
what are the dimensions of surface tension
samsfavor
why is the "_" sign used for a wave to the right instead of to the left?
why classical mechanics is necessary for graduate students?
classical mechanics?
Victor
principle of superposition?
principle of superposition allows us to find the electric field on a charge by finding the x and y components
Kidus
Two Masses,m and 2m,approach each along a path at right angles to each other .After collision,they stick together and move off at 2m/s at angle 37° to the original direction of the mass m. What where the initial speeds of the two particles
MB
2m & m initial velocity 1.8m/s & 4.8m/s respectively,apply conservation of linear momentum in two perpendicular directions.
Shubhrant
A body on circular orbit makes an angular displacement given by teta(t)=2(t)+5(t)+5.if time t is in seconds calculate the angular velocity at t=2s
MB
2+5+0=7sec differentiate above equation w.r.t time, as angular velocity is rate of change of angular displacement.
Shubhrant
Ok i got a question I'm not asking how gravity works. I would like to know why gravity works. like why is gravity the way it is. What is the true nature of gravity?
gravity pulls towards a mass...like every object is pulled towards earth
Ashok
An automobile traveling with an initial velocity of 25m/s is accelerated to 35m/s in 6s,the wheel of the automobile is 80cm in diameter. find * The angular acceleration
(10/6) ÷0.4=4.167 per sec
Shubhrant
what is the formula for pressure?
force/area
Kidus
force is newtom
Kidus
and area is meter squared
Kidus
so in SI units pressure is N/m^2
Kidus
In customary United States units pressure is lb/in^2. pound per square inch
Kidus
who is Newton?
scientist
Jeevan
a scientist
Peter
that discovered law of motion
Peter
ok
John
but who is Isaac newton?
John
a postmodernist would say that he did not discover them, he made them up and they're not actually a reality in itself, but a mere construct by which we decided to observe the word around us
elo
how?
Qhoshe
Besides his work on universal gravitation (gravity), Newton developed the 3 laws of motion which form the basic principles of modern physics. His discovery of calculus led the way to more powerful methods of solving mathematical problems. His work in optics included the study of white light and
Daniel
and the color spectrum
Daniel