# 10.2 Kinematics of rotational motion

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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 $\theta$ , $\omega$ , and $\alpha$ 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 $\alpha$ is large for a long period of time $t$ , then the final angular velocity $\omega$ and angle of rotation $\theta$ 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 $\omega$ , $\alpha$ , and $t$ . To determine this equation, we recall a familiar kinematic equation for translational, or straight-line, motion:

Note that in rotational motion $a={a}_{\text{t}}$ , and we shall use the symbol $a$ for tangential or linear acceleration from now on. As in linear kinematics, we assume $a$ is constant, which means that angular acceleration $\alpha$ is also a constant, because $a=\mathrm{r\alpha }$ . Now, let us substitute $v=\mathrm{r\omega }$ and $a=\mathrm{r\alpha }$ into the linear equation above:

$\mathrm{r\omega }={\mathrm{r\omega }}_{0}+\mathrm{r\alpha t}.$

The radius $r$ cancels in the equation, yielding

where ${\omega }_{0}$ is the initial angular velocity. This last equation is a kinematic relationship among $\omega$ , $\alpha$ , and $t$ —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
$\theta =\overline{\omega }t$ $x=\stackrel{-}{v}t$
$\omega ={\omega }_{0}+\mathrm{\alpha t}$ $v={v}_{0}+\text{at}$ (constant $\alpha$ , $a$ )
$\theta ={\omega }_{0}t+\frac{1}{2}{\mathrm{\alpha t}}^{2}$ $x={v}_{0}t+\frac{1}{2}{\text{at}}^{2}$ (constant $\alpha$ , $a$ )
${\omega }^{2}={{\omega }_{0}}^{2}+2\text{αθ}$ ${v}^{2}={{v}_{0}}^{2}+2\text{ax}$ (constant $\alpha$ , $a$ )

In these equations, the subscript 0 denotes initial values ( ${\theta }_{0}$ , ${x}_{0}$ , and ${t}_{0}$ are initial values), and the average angular velocity $\stackrel{-}{\omega }$ and average velocity $\stackrel{-}{v}$ are defined as follows:

The equations given above in [link] can be used to solve any rotational or translational kinematics problem in which $a$ and $\alpha$ 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
how
Bernard
that resistance is not measured yet, it may be probably in the next generation of scientists
Paul
Is fundamental quantities under physical quantities?
please I didn't not understand the concept of the physical therapy
physiotherapy - it's a practice of exercising for healthy living.
Paul
what chapter is this?
Anderson
please I have probably with calculate please can you please and help me out
Sure
What is Boyce law
how to convert meter per second to kilometers per hour
Divide with 3.6
Mateo
multiply by (km/1000m) x (3600 s/h) -> 3.6
2 how heat loss is prevented in a vacuum flask
what is science
Helen
logical reasoning for a particular phenomenon.
Ajay
I don't know anything about it 😔. I'm sorry, please forgive 😔
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
Abdul
dimensions v²=u²+2at
what if time is not given in finding the average velocity?
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..
How do you calculate precision
what module is that?
Fillemon
Chemisty 1A?
Fillemon
No it has something to do with measurements bro... What we did today in class
Sacky
Tah bra honestly I didn't understand a thing in that class..when re your Tutorials?
Fillemon
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
Sacky
I eat ass
Anderson
I'll work on the maths sheet tomorrow bra @Sacky Malyenge but I'll try mastering them
Fillemon
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Anderson
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Anderson
lol, hush
Emi
There are very large numbers of charged particles in most objects. Why, then, don’t most objects exhibit static electricity?
Because there's an equal number of negative and positive charges... objects are neutral in nature
NELSON
when a ball rolls on a smooth level ground,the motion of its centre is?
what is electro magnetic field?
Mary
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..!!!
NELSON
Electromagnetic field is caused by moving electric charge
when a ball rolls on a smooth level ground,the motion of its centre is?
Mumeh
what's the relationship btw displacement and position
displacement is the change of position 8======✊=D 💦💦
Anderson
what is the meaning of elasticity
is the ability of a material to or any object to expand to a limit point
king
this is about kinematics you bonk
Emi