<< Chapter < Page Chapter >> Page >
  • Describe uniform circular motion.
  • Explain non-uniform circular motion.
  • Calculate angular acceleration of an object.
  • Observe the link between linear and angular acceleration.

Uniform Circular Motion and Gravitation discussed only uniform circular motion, which is motion in a circle at constant speed and, hence, constant angular velocity. Recall that angular velocity ω size 12{ω} {} was defined as the time rate of change of angle θ size 12{θ} {} :

ω = Δ θ Δ t , size 12{ω= { {Δθ} over {Δt} } ","} {}

where θ size 12{θ} {} is the angle of rotation as seen in [link] . The relationship between angular velocity ω size 12{ω} {} and linear velocity v size 12{v} {} was also defined in Rotation Angle and Angular Velocity as

v = size 12{v=rω} {}

or

ω = v r , size 12{ω= { {v} over {r} } } {}

where r size 12{r} {} is the radius of curvature, also seen in [link] . According to the sign convention, the counter clockwise direction is considered as positive direction and clockwise direction as negative

The given figure shows counterclockwise circular motion with a horizontal line, depicting radius r, drawn from the center of the circle to the right side on its circumference and another line is drawn in such a manner that it makes an acute angle delta theta with the horizontal line. Tangential velocity vectors are indicated at the end of the two lines. At the bottom right side of the figure, the formula for angular velocity is given as v upon r.
This figure shows uniform circular motion and some of its defined quantities.

Angular velocity is not constant when a skater pulls in her arms, when a child starts up a merry-go-round from rest, or when a computer’s hard disk slows to a halt when switched off. In all these cases, there is an angular acceleration    , in which ω size 12{ω} {} changes. The faster the change occurs, the greater the angular acceleration. Angular acceleration α size 12{α} {} is defined as the rate of change of angular velocity. In equation form, angular acceleration is expressed as follows:

α = Δ ω Δ t , size 12{α= { {Δω} over {Δt} } ","} {}

where Δ ω size 12{Δω} {} is the change in angular velocity    and Δ t size 12{Δt} {} is the change in time. The units of angular acceleration are rad/s /s size 12{ left ("rad/s" right )"/s"} {} , or rad/s 2 size 12{"rad/s" rSup { size 8{2} } } {} . If ω size 12{ω} {} increases, then α size 12{α} {} is positive. If ω size 12{ω} {} decreases, then α size 12{α} {} is negative.

Calculating the angular acceleration and deceleration of a bike wheel

Suppose a teenager puts her bicycle on its back and starts the rear wheel spinning from rest to a final angular velocity of 250 rpm in 5.00 s. (a) Calculate the angular acceleration in rad/s 2 size 12{"rad/s" rSup { size 8{2} } } {} . (b) If she now slams on the brakes, causing an angular acceleration of 87.3 rad/s 2 size 12{"-87" "." 3`"rad/s" rSup { size 8{2} } } {} , how long does it take the wheel to stop?

Strategy for (a)

The angular acceleration can be found directly from its definition in α = Δ ω Δ t size 12{α= { {Δω} over {Δt} } } {} because the final angular velocity and time are given. We see that Δ ω size 12{Δω} {} is 250 rpm and Δ t size 12{Δt} {} is 5.00 s.

Solution for (a)

Entering known information into the definition of angular acceleration, we get

α = Δ ω Δ t = 250 rpm 5.00 s . alignl { stack { size 12{α= { {Δω} over {Δt} } } {} #size 12{ {}= { {"250"" rpm"} over {5 "." "00 s"} } "."} {} } } {}

Because Δ ω size 12{Δω} {} is in revolutions per minute (rpm) and we want the standard units of rad/s 2 size 12{"rad/s" rSup { size 8{2} } } {} for angular acceleration, we need to convert Δ ω size 12{Δω} {} from rpm to rad/s:

Δ ω = 250 rev min 2π rad rev 1 min 60 sec = 26.2 rad s . alignl { stack { size 12{Δω="250" { {"rev"} over {"min"} } cdot { {2π" rad"} over {"60" "." "0 s"} } } {} #size 12{ {}="26" "." 2 { {"rad"} over {"s"} } } {} } } {}

Entering this quantity into the expression for α size 12{α} {} , we get

α = Δ ω Δ t = 26.2 rad/s 5.00 s = 5.24  rad/s 2 . alignl { stack { size 12{α= { {Δω} over {Δt} } } {} #size 12{ {}= { {"26" "." 2" rad/s"} over {5 "." "00"" s"} } "." } {} # size 12{ {}=5 "." "24"" rad/s" rSup { size 8{2} } } {}} } {}

Strategy for (b)

In this part, we know the angular acceleration and the initial angular velocity. We can find the stoppage time by using the definition of angular acceleration and solving for Δ t size 12{Δt} {} , yielding

Δ t = Δ ω α . size 12{Δt= { {Δω} over {α} } "."} {}

Solution for (b)

Here the angular velocity decreases from 26.2 rad/s size 12{"26" "." 2`"rad/s"} {} (250 rpm) to zero, so that Δ ω size 12{Δω} {} is 26.2 rad/s , and α size 12{α} {} is given to be 87.3 rad/s 2 size 12{"-87" "." 3`"rad/s" rSup { size 8{2} } } {} . Thus,

Δ t = 26.2 rad/s 87.3 rad/s 2 = 0.300 s. alignl { stack { size 12{Δt= { { - "26" "." 2`"rad/s"} over { - "87" "." 3`"rad/s" rSup { size 8{2} } } } } {} #=0 "." "300"`"s" "." {} } } {}

Discussion

Note that the angular acceleration as the girl spins the wheel is small and positive; it takes 5 s to produce an appreciable angular velocity. When she hits the brake, the angular acceleration is large and negative. The angular velocity quickly goes to zero. In both cases, the relationships are analogous to what happens with linear motion. For example, there is a large deceleration when you crash into a brick wall—the velocity change is large in a short time interval.

Questions & Answers

I'm interested in biological psychology and cognitive psychology
Tanya Reply
what does preconceived mean
sammie Reply
physiological Psychology
Nwosu Reply
How can I develope my cognitive domain
Amanyire Reply
why is communication effective
Dakolo Reply
Communication is effective because it allows individuals to share ideas, thoughts, and information with others.
effective communication can lead to improved outcomes in various settings, including personal relationships, business environments, and educational settings. By communicating effectively, individuals can negotiate effectively, solve problems collaboratively, and work towards common goals.
it starts up serve and return practice/assessments.it helps find voice talking therapy also assessments through relaxed conversation.
miss
Every time someone flushes a toilet in the apartment building, the person begins to jumb back automatically after hearing the flush, before the water temperature changes. Identify the types of learning, if it is classical conditioning identify the NS, UCS, CS and CR. If it is operant conditioning, identify the type of consequence positive reinforcement, negative reinforcement or punishment
Wekolamo Reply
please i need answer
Wekolamo
because it helps many people around the world to understand how to interact with other people and understand them well, for example at work (job).
Manix Reply
Agreed 👍 There are many parts of our brains and behaviors, we really need to get to know. Blessings for everyone and happy Sunday!
ARC
A child is a member of community not society elucidate ?
JESSY Reply
Isn't practices worldwide, be it psychology, be it science. isn't much just a false belief of control over something the mind cannot truly comprehend?
Simon Reply
compare and contrast skinner's perspective on personality development on freud
namakula Reply
Skinner skipped the whole unconscious phenomenon and rather emphasized on classical conditioning
war
explain how nature and nurture affect the development and later the productivity of an individual.
Amesalu Reply
nature is an hereditary factor while nurture is an environmental factor which constitute an individual personality. so if an individual's parent has a deviant behavior and was also brought up in an deviant environment, observation of the behavior and the inborn trait we make the individual deviant.
Samuel
I am taking this course because I am hoping that I could somehow learn more about my chosen field of interest and due to the fact that being a PsyD really ignites my passion as an individual the more I hope to learn about developing and literally explore the complexity of my critical thinking skills
Zyryn Reply
good👍
Jonathan
and having a good philosophy of the world is like a sandwich and a peanut butter 👍
Jonathan
generally amnesi how long yrs memory loss
Kelu Reply
interpersonal relationships
Abdulfatai Reply
What would be the best educational aid(s) for gifted kids/savants?
Heidi Reply
treat them normal, if they want help then give them. that will make everyone happy
Saurabh
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 3

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College physics arranged for cpslo phys141. OpenStax CNX. Dec 23, 2014 Download for free at http://legacy.cnx.org/content/col11718/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College physics arranged for cpslo phys141' conversation and receive update notifications?

Ask