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  • Observe the kinematics of rotational motion.
  • Derive rotational kinematic equations.
  • Evaluate problem solving strategies for rotational kinematics.

Just by using our intuition, we can begin to see how rotational quantities like θ size 12{θ} {} , ω size 12{ω} {} , and α size 12{α} {} are related to one another. For example, if a motorcycle wheel has a large angular acceleration for a fairly long time, it ends up spinning rapidly and rotates through many revolutions. In more technical terms, if the wheel’s angular acceleration α size 12{α} {} is large for a long period of time t size 12{α} {} , then the final angular velocity ω size 12{ω} {} and angle of rotation θ size 12{θ} {} are large. The wheel’s rotational motion is exactly analogous to the fact that the motorcycle’s large translational acceleration produces a large final velocity, and the distance traveled will also be large.

Kinematics is the description of motion. The kinematics of rotational motion    describes the relationships among rotation angle, angular velocity, angular acceleration, and time. Let us start by finding an equation relating ω size 12{ω} {} , α size 12{α} {} , and t size 12{t} {} . To determine this equation, we recall a familiar kinematic equation for translational, or straight-line, motion:

v = v 0 + at       ( constant  a ) size 12{v=v rSub { size 8{0} } + ital "at"" " \[ "constant "a \] } {}

Note that in rotational motion a = a t size 12{a=a rSub { size 8{t} } } {} , and we shall use the symbol a size 12{a} {} for tangential or linear acceleration from now on. As in linear kinematics, we assume a size 12{a} {} is constant, which means that angular acceleration α size 12{α} {} is also a constant, because a = size 12{a=rα} {} . Now, let us substitute v = size 12{v=rω} {} and a = size 12{a=rα} {} into the linear equation above:

= 0 + rαt . size 12{rω=rω rSub { size 8{0} } +rαt} {}

The radius r size 12{r} {} cancels in the equation, yielding

ω = ω 0 + at       ( constant  a ) , size 12{ω=ω rSub { size 8{0} } + ital "at"" " \[ "constant "a \] ,} {}

where ω 0 size 12{ω rSub { size 8{0} } } {} is the initial angular velocity. This last equation is a kinematic relationship among ω size 12{ω} {} , α size 12{α} {} , and t size 12{t} {} —that is, it describes their relationship without reference to forces or masses that may affect rotation. It is also precisely analogous in form to its translational counterpart.

Making connections

Kinematics for rotational motion is completely analogous to translational kinematics, first presented in One-Dimensional Kinematics . Kinematics is concerned with the description of motion without regard to force or mass. We will find that translational kinematic quantities, such as displacement, velocity, and acceleration have direct analogs in rotational motion.

Starting with the four kinematic equations we developed in One-Dimensional Kinematics , we can derive the following four rotational kinematic equations (presented together with their translational counterparts):

Rotational kinematic equations
Rotational Translational
θ = ω ¯ t size 12{θ= {overline {ωt}} } {} x = v - t size 12{x= { bar {v}}t} {}
ω = ω 0 + αt size 12{ω=ω rSub { size 8{0} } +αt} {} v = v 0 + at size 12{v=v rSub { size 8{0} } + ital "at"} {} (constant α size 12{α} {} , a size 12{a} {} )
θ = ω 0 t + 1 2 αt 2 size 12{θ=ω rSub { size 8{0} } t+ { {1} over {2} } αt rSup { size 8{2} } } {} x = v 0 t + 1 2 at 2 size 12{x=v rSub { size 8{0} } t+ { {1} over {2} } ital "at" rSup { size 8{2} } } {} (constant α size 12{α} {} , a size 12{a} {} )
ω 2 = ω 0 2 + 2 αθ size 12{ω rSup { size 8{2} } =ω rSub { size 8{0} rSup { size 8{2} } } +2 ital "αθ"} {} v 2 = v 0 2 + 2 ax (constant α , a )

In these equations, the subscript 0 denotes initial values ( θ 0 size 12{θ rSub { size 8{0} } } {} , x 0 size 12{x rSub { size 8{0} } } {} , and t 0 size 12{t rSub { size 8{0} } } {} are initial values), and the average angular velocity ω - size 12{ { bar {ω}}} {} and average velocity v - size 12{ { bar {v}}} {} are defined as follows:

ω ¯ = ω 0 + ω 2  and  v ¯ = v 0 + v 2 . size 12{ {overline {ω}} = { {ω rSub { size 8{0} } +ω} over {2} } " and " {overline {v}} = { {v rSub { size 8{0} } +v} over {2} } " " \( "constant "α, a \) } {}

The equations given above in [link] can be used to solve any rotational or translational kinematics problem in which a size 12{a} {} and α size 12{α} {} are constant.

Problem-solving strategy for rotational kinematics

  1. Examine the situation to determine that rotational kinematics (rotational motion) is involved . Rotation must be involved, but without the need to consider forces or masses that affect the motion.
  2. Identify exactly what needs to be determined in the problem (identify the unknowns) . A sketch of the situation is useful.
  3. Make a list of what is given or can be inferred from the problem as stated (identify the knowns) .
  4. Solve the appropriate equation or equations for the quantity to be determined (the unknown) . It can be useful to think in terms of a translational analog because by now you are familiar with such motion.
  5. Substitute the known values along with their units into the appropriate equation, and obtain numerical solutions complete with units . Be sure to use units of radians for angles.
  6. Check your answer to see if it is reasonable: Does your answer make sense ?

Questions & Answers

resistance of thermometer in relation to temperature
Ifeanyi Reply
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Igwe Reply
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John Reply
What is Boyce law
Sly Reply
how to convert meter per second to kilometers per hour
grace Reply
Divide with 3.6
multiply by (km/1000m) x (3600 s/h) -> 3.6
2 how heat loss is prevented in a vacuum flask
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logical reasoning for a particular phenomenon.
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due to non in contact mean no conduction and no convection bec of non conducting base and walls and also their is a grape between the layer like to take the example of thermo flask
dimensions v²=u²+2at
Lagben Reply
what if time is not given in finding the average velocity?
Alan Reply
the magnetic circuit of a certain of the flux paths in each of the long and short sides being 25cm and 20cm reprectielectrove. there is an air gap of 2mm long in one the long sides if a flux density of 0.8weber/m is to produce in the magnet of 1500 turns..
Daniel Reply
How do you calculate precision
Sacky Reply
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Chemisty 1A?
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Tah bra honestly I didn't understand a thing in that class..when re your Tutorials?
Friday bro... But the topics we did are in this app... Just try to master them quickly before the test dates... Are you done with the Maths sheet
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There are very large numbers of charged particles in most objects. Why, then, don’t most objects exhibit static electricity?
Bilkisu Reply
Because there's an equal number of negative and positive charges... objects are neutral in nature
when a ball rolls on a smooth level ground,the motion of its centre is?
Mary Reply
what is electro magnetic field?
electromagnetic field is a special type of field been produced by electric charges..!!! like the word electro from Electricity and the word magnetic from Magnetism.. so it is more of a join field..!!!
Electromagnetic field is caused by moving electric charge
when a ball rolls on a smooth level ground,the motion of its centre is?
what's the relationship btw displacement and position
Declan Reply
displacement is the change of position 8======✊=D 💦💦
what is the meaning of elasticity
Pele Reply
is the ability of a material to or any object to expand to a limit point
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Practice Key Terms 1

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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