# 10.1 Rotational variables  (Page 4/11)

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$\alpha =\underset{\text{Δ}t\to 0}{\text{lim}}\frac{\text{Δ}\omega }{\text{Δ}t}=\frac{d\omega }{dt}=\frac{{d}^{2}\theta }{d{t}^{2}},$

where we have taken the limit of the average angular acceleration, $\stackrel{–}{\alpha }=\frac{\text{Δ}\omega }{\text{Δ}t}$ as $\text{Δ}t\to 0$ .

The units of angular acceleration are (rad/s)/s, or ${\text{rad/s}}^{2}$ .

In the same way as we defined the vector associated with angular velocity $\stackrel{\to }{\omega }$ , we can define $\stackrel{\to }{\alpha }$ , the vector associated with angular acceleration ( [link] ). If the angular velocity is along the positive z- axis, as in [link] , and $\frac{d\omega }{dt}$ is positive, then the angular acceleration $\stackrel{\to }{\alpha }$ is positive and points along the $+z\text{-}$ axis. Similarly, if the angular velocity $\stackrel{\to }{\omega }$ is along the positive z- axis and $\frac{d\omega }{dt}$ is negative, then the angular acceleration is negative and points along the $+z-$ axis. The rotation is counterclockwise in both (a) and (b) with the angular velocity in the same direction. (a) The angular acceleration is in the same direction as the angular velocity, which increases the rotation rate. (b) The angular acceleration is in the opposite direction to the angular velocity, which decreases the rotation rate.

We can express the tangential acceleration vector as a cross product of the angular acceleration and the position vector. This expression can be found by taking the time derivative of $\stackrel{\to }{v}=\stackrel{\to }{\omega }\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{r}$ and is left as an exercise:

$\stackrel{\to }{\mathbf{\text{a}}}=\stackrel{\to }{\mathbf{\text{α}}}\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}\stackrel{\to }{\mathbf{\text{r}}}.$

The vector relationships for the angular acceleration and tangential acceleration are shown in [link] . (a) The angular acceleration is the positive z -direction and produces a tangential acceleration in a counterclockwise sense. (b) The angular acceleration is in the negative z -direction and produces a tangential acceleration in the clockwise sense.

We can relate the tangential acceleration of a point on a rotating body at a distance from the axis of rotation in the same way that we related the tangential speed to the angular velocity. If we differentiate [link] with respect to time, noting that the radius r is constant, we obtain

${a}_{\text{t}}=r\alpha .$

Thus, the tangential acceleration ${a}_{t}$ is the radius times the angular acceleration. [link] and [link] are important for the discussion of rolling motion (see Angular Momentum ).

Let’s apply these ideas to the analysis of a few simple fixed-axis rotation scenarios. Before doing so, we present a problem-solving strategy that can be applied to rotational kinematics: the description of rotational motion.

## Problem-solving strategy: rotational kinematics

1. Examine the situation to determine that rotational kinematics (rotational motion) is involved.
2. Identify exactly what needs to be determined in the problem (identify the unknowns). A sketch of the situation is useful.
3. Make a complete 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 the equations of translational 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.

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refer to how fast an object rotates ot revolve to another point
samson
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x=x0+v0t+1/2at^2
Grant
what is the deference between precision and accuracy
ShAmy
In measurement of a set, accuracy refers to closeness of the measurements to a specific value, while precision refers to the closeness of the measurements to each other.
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Thank you Mr..
ShAmy
Are there is a difference between Error and Relative Error...
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670.8km
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which formula did you use
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X^2=300^2+600^2
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impulse by height fomula
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magnitude and direction
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what is impulse?
the product of force and time
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friend ,what is the equation your question
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p=mv
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what is the formula for angular displacement ?
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angular displacement = l/r
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friend ,what is the " l "?
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I have another problem,what is the difference between pure chemistry and applied chemistry ?
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l stands for linear displacement and l>>r
anand
r stands for radius or position vector of particle
anand
pure chemistry is related to classical and basic concepts of chemistry while applied Chemistry deals with it's application oriented concepts and procedures.
anand
what is the deference between precision and accuracy?
ShAmy
I think in general both are same , more accuracy means more precise
anand
and lessor error
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anand,can you give an example for angular displacement ?
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when any particle rotates or revolves about something , an axis or point ,then angle traversed is angular displacement
anand
like a revolving fan
anand
yeah ,I understood now, thanks!
lasitha
o need this application in French language
Why is it that the definition is not there
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the study of matter in relation to energy
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Ampher law
Zia By By  By Rhodes    By Rhodes  