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So far, we have defined three rotational quantities— θ ω size 12{θ,ω} {} , and α size 12{α} {} . These quantities are analogous to the translational quantities x v size 12{x,v} {} , and a size 12{a} {} . [link] displays rotational quantities, the analogous translational quantities, and the relationships between them.

Rotational and translational quantities
Rotational Translational Relationship
θ size 12{θ} {} x size 12{x} {} θ = x r size 12{θ= { {x} over {r} } } {}
ω size 12{ω} {} v size 12{v} {} ω = v r size 12{ω= { {v} over {r} } } {}
α size 12{α} {} a size 12{a} {} α = a t r size 12{α= { {a rSub { size 8{t} } } over {r} } } {}

Making connections: take-home experiment

Sit down with your feet on the ground on a chair that rotates. Lift one of your legs such that it is unbent (straightened out). Using the other leg, begin to rotate yourself by pushing on the ground. Stop using your leg to push the ground but allow the chair to rotate. From the origin where you began, sketch the angle, angular velocity, and angular acceleration of your leg as a function of time in the form of three separate graphs. Estimate the magnitudes of these quantities.

Angular acceleration is a vector, having both magnitude and direction. How do we denote its magnitude and direction? Illustrate with an example.

The magnitude of angular acceleration is α size 12{α} {} and its most common units are rad/s 2 size 12{"rad/s" rSup { size 8{2} } } {} . The direction of angular acceleration along a fixed axis is denoted by a + or a – sign, just as the direction of linear acceleration in one dimension is denoted by a + or a – sign. For example, consider a gymnast doing a forward flip. Her angular momentum would be parallel to the mat and to her left. The magnitude of her angular acceleration would be proportional to her angular velocity (spin rate) and her moment of inertia about her spin axis.

Phet explorations: ladybug revolution

Join the ladybug in an exploration of rotational motion. Rotate the merry-go-round to change its angle, or choose a constant angular velocity or angular acceleration. Explore how circular motion relates to the bug's x,y position, velocity, and acceleration using vectors or graphs.

Ladybug Revolution

Section summary

  • Uniform circular motion is the motion with a constant angular velocity ω = Δ θ Δ t size 12{ω= { {Δθ} over {Δt} } } {} .
  • In non-uniform circular motion, the velocity changes with time and the rate of change of angular velocity (i.e. angular acceleration) is α = Δ ω Δ t size 12{α= { {Δω} over {Δt} } } {} .
  • Linear or tangential acceleration refers to changes in the magnitude of velocity but not its direction, given as a t = Δ v Δ t size 12{a rSub { size 8{t} } = { {Δv} over {Δt} } } {} .
  • For circular motion, note that v = size 12{v=rω} {} , so that
    a t = Δ Δ t . size 12{a rSub { size 8{t} } = { {Δ left (rω right )} over {Δt} } } {}
  • The radius r is constant for circular motion, and so Δ = r Δ ω size 12{Δ left (rω right )=rΔω} {} . Thus,
    a t = r Δ ω Δ t . size 12{a rSub { size 8{t} } =r { {Δω} over {Δt} } } {}
  • By definition, Δ ω / Δ t = α size 12{ {Δω} slash {Δt=α} } {} . Thus,
    a t = size 12{a rSub { size 8{t} } =rα} {}

    or

    α = a t r . size 12{α= { {a rSub { size 8{t} } } over {r} } } {}

Conceptual questions

Analogies exist between rotational and translational physical quantities. Identify the rotational term analogous to each of the following: acceleration, force, mass, work, translational kinetic energy, linear momentum, impulse.

Explain why centripetal acceleration changes the direction of velocity in circular motion but not its magnitude.

In circular motion, a tangential acceleration can change the magnitude of the velocity but not its direction. Explain your answer.

Suppose a piece of food is on the edge of a rotating microwave oven plate. Does it experience nonzero tangential acceleration, centripetal acceleration, or both when: (a) The plate starts to spin? (b) The plate rotates at constant angular velocity? (c) The plate slows to a halt?

Problems&Exercises

At its peak, a tornado is 60.0 m in diameter and carries 500 km/h winds. What is its angular velocity in revolutions per second?

ω = 0 . 737 rev/s size 12{ω= {underline {0 "." "737 rev/s"}} } {}

Integrated Concepts

An ultracentrifuge accelerates from rest to 100,000 rpm in 2.00 min. (a) What is its angular acceleration in rad/s 2 size 12{"rad/s" rSup { size 8{2} } } {} ? (b) What is the tangential acceleration of a point 9.50 cm from the axis of rotation? (c) What is the radial acceleration in m/s 2 size 12{"m/s" rSup { size 8{2} } } {} and multiples of g size 12{gs} {} of this point at full rpm?

Integrated Concepts

You have a grindstone (a disk) that is 90.0 kg, has a 0.340-m radius, and is turning at 90.0 rpm, and you press a steel axe against it with a radial force of 20.0 N. (a) Assuming the kinetic coefficient of friction between steel and stone is 0.20, calculate the angular acceleration of the grindstone. (b) How many turns will the stone make before coming to rest?

(a) 0 . 26 rad/s 2 size 12{ - 0 "." "26 rad/s" rSup { size 8{2} } } {}

(b) 27 rev size 12{"27"`"rev"} {}

Unreasonable Results

You are told that a basketball player spins the ball with an angular acceleration of 100  rad/s 2 size 12{"100"``"rad/s" rSup { size 8{2} } } {} . (a) What is the ball’s final angular velocity if the ball starts from rest and the acceleration lasts 2.00 s? (b) What is unreasonable about the result? (c) Which premises are unreasonable or inconsistent?

Questions & Answers

I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Introduction to applied math and physics. OpenStax CNX. Oct 04, 2012 Download for free at http://cnx.org/content/col11426/1.3
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