10.1 Rotational variables (Page 6/11)

University physics volume 1 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 $θ$ 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 $ω (rad / s)$ , the rotational rate of the body in radians per second. The instantaneous angular velocity of a rotating body $ω = lim_{Δ t \to 0} \frac{Δ ω}{Δ t} = \frac{d θ}{d t}$ is the derivative with respect to time of the angular position $θ$ , found by taking the limit $Δ t \to 0$ in the average angular velocity $\bar{ω} = \frac{Δ θ}{Δ t}$ . The angular velocity relates $v_{t}$ to the tangential speed of a point on the rotating body through the relation $v_{t} = r ω$ , where r is the radius to the point and $v_{t}$ is the tangential speed at the given point.
The angular velocity $\vec{ω}$ 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 $\vec{ω}$ (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, $\bar{α} = \frac{Δ ω}{Δ t}$ . The instantaneous angular acceleration is the time derivative of angular velocity, $α = lim_{Δ t \to 0} \frac{Δ ω}{Δ t} = \frac{d ω}{d t}$ . The angular acceleration $\vec{α}$ 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 $\vec{ω}$ . If the rotation rate is increasing, the angular acceleration is in the same direction as $\vec{ω}$ .
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.

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!

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?

Got questions? Get instant answers now!

Problems

Calculate the angular velocity of Earth.

Got questions? Get instant answers now!

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?

$ω = \frac{2 π rad}{45.0 s} = 0.14 rad/s$

Got questions? Get instant answers now!

A wheel rotates at a constant rate of $2.0 \times 10^{3} rev / min$ . (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.

Got questions? Get instant answers now!

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. $θ = \frac{s}{r} = \frac{3.0 m}{1.5 m} = 2.0 rad$ ; b. $ω = \frac{2.0 rad}{1.0 s} = 2.0 rad/s$ ; c. $\frac{v^{2}}{r} = \frac{{(3.0 m/s)}^{2}}{1.5 m} = 6.0 m / s^{2} .$

Got questions? Get instant answers now!

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?

Got questions? Get instant answers now!

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 $2.0 rad / s^{2}$ 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 $Δ t = \frac{Δ ω}{α} = \frac{0 rad/s - 10.0 (2 π) rad/s}{−2.0 rad / s^{2}} = 31.4 s$ 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.

Got questions? Get instant answers now!

A gyroscope slows from an initial rate of 32.0 rad/s at a rate of $0.700 {rad/s}^{2}$ . How long does it take to come to rest?

Got questions? Get instant answers now!

On takeoff, the propellers on a UAV (unmanned aerial vehicle) increase their angular velocity from rest at a rate of $ω = (25.0 t) rad / s$ for 3.0 s. (a) What is the instantaneous angular velocity of the propellers at $t = 2.0 s$ ? (b) What is the angular acceleration?

a. $ω = 25.0 (2.0 s) = 50.0 rad/s$ ; b. $α = \frac{d ω}{d t} = 25.0 {rad/s}^{2}$

Got questions? Get instant answers now!

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 $t = 5 s ?$ (b) What is the angular acceleration of the rod? (c) What are the tangential speeds of the beads at $t = 5 s ?$ (d) What are the tangential accelerations of the beads at $t = 5 s ?$ (e) What are the centripetal accelerations of the beads at $t = 5 s ?$

Figure is a drawing of a rod that rotates counterclockwise. Rod has two beads on it, one at 10 cm from the rotation axis and the other at 20 cm from the rotation axis.

Got questions? Get instant answers now!

Questions & Answers

prostaglandin and fever

Maha Reply

Discuss the differences between taste and flavor, including how other sensory inputs contribute to our perception of flavor.

John Reply

taste refers to your understanding of the flavor . while flavor one The other hand is refers to sort of just a blend things.

Faith

While taste primarily relies on our taste buds, flavor involves a complex interplay between taste and aroma

Kamara

which drugs can we use for ulcers

Ummi Reply

omeprazole

Kamara

what

Renee

what is this

Renee

is a drug

Kamara

of anti-ulcer

Kamara

Omeprazole Cimetidine / Tagament For the complicated once ulcer - kit

Patrick

what is the function of lymphatic system

Nency Reply

Not really sure

Eli

to drain extracellular fluid all over the body.

asegid

The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include: 1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of

asegid

to transport fluids fats proteins and lymphocytes to the blood stream as lymph

Adama

what is anatomy

Oyindarmola Reply

Anatomy is the identification and description of the structures of living things

Kamara

what's the difference between anatomy and physiology

Oyerinde Reply

Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.

AI-Robot

what is enzymes all about?

Mohammed Reply

Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems

Kamara

yes

Prince

how does the stomach protect itself from the damaging effects of HCl

Wulku Reply

little girl okay how does the stomach protect itself from the damaging effect of HCL

Wulku

it is because of the enzyme that the stomach produce that help the stomach from the damaging effect of HCL

Kamara

function of digestive system

Ali Reply

function of digestive

Ali

the diagram of the lungs

Adaeze Reply

what is the normal body temperature

Diya Reply

37 degrees selcius

Xolo

37°c

Stephanie

please why 37 degree selcius normal temperature

Mark

36.5

Simon

37°c

Iyogho

the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature

Stephanie

37A c

Wulku

what is anaemia

Diya Reply

anaemia is the decrease in RBC count hemoglobin count and PVC count

Eniola

what is the pH of the vagina

Diya Reply

how does Lysin attack pathogens

Diya

acid

Mary

I information on anatomy position and digestive system and there enzyme

Elisha Reply

anatomy of the female external genitalia

Muhammad Reply

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Practice Key Terms 5

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'University physics volume 1' conversation and receive update notifications?

Ask

	28 AP 28 Development Inheritance Essay By OpenStax Start Flashcards
©flickr: BK	Self Confidence By Miranda Reising Start Quiz
	Macroeconomics By OpenStax Read Online Course
©flickr: Dutch	Las Vegas Timeshare By Donyea Sweets Start Test
	1 Neuroanatomy 01 The Cranial Nerves By Stephen Voron Start Quiz
	2 AP 02 Chemical Level of Organization MCQ By OpenStax Start Quiz
	11 Clin Sci I - GI Twedt quiz 2 By Brooke Delaney Start Exam
©flickr: Eduardo	Nursing Education NU589 Program Outcomes By Karen Gowdey Start Quiz
	19 Sociology 19 Health and Medicine MCQ By OpenStax Start Quiz
	10 Biology 10 Cell Reproduction MCQ By OpenStax Start Quiz