Kinematics of Rotational Motion
Learning Objectives
By the end of this section, you will be able to: Observe the kinematics of rotational motion.
 Derive rotational kinematic equations.
 Evaluate problem solving strategies for rotational kinematics.
rω = rω_{0} + rat.
The radius r cancels in the equation, yieldingω = ω_{0} + at. (constant a)
where ω_{0} is the initial angular velocity. This last equation is a kinematic relationship among ω, α, 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
Rotational  Translational  

$\theta =\bar{\omega }t\\$ 
$x=\bar{v}t\\$ 

ω = ω_{0 } + αt  v = v_{o} + at  (constant α, a) 
$\theta ={\omega }_{0}t+\frac{1}{2}{\alpha t}^{2}\\$ 
$x={v}_{0}t+\frac{1}{2}{\text{at}}^{2}\\$ 
(constant α, a) 
ω^{2} = ω_{0}^{2}+ 2αθ  v^{2} = v_{o}^{2} + 2ax  (constant α, a) 
ProblemSolving Strategy for Rotational Kinematics
 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.
 Identify exactly what needs to be determined in the problem (identify the unknowns). A sketch of the situation is useful.
 Make a list of what is given or can be inferred from the problem as stated (identify the knowns).
 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.
 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.
 Check your answer to see if it is reasonable: Does your answer make sense?
Example 1. Calculating the Acceleration of a Fishing Reel
Strategy
In each part of this example, the strategy is the same as it was for solving problems in linear kinematics. In particular, known values are identified and a relationship is then sought that can be used to solve for the unknown.Solution for (a)
Here α and t are given and ω needs to be determined. The most straightforward equation to use is ω = ω_{0}+αt because the unknown is already on one side and all other terms are known. That equation states thatω = 0 + (110 rad/s^{2})(2.00 s) = 220 rad/s
Solution for (b)
Now that ω is known, the speed v can most easily be found using the relationshipSolution for (c)
Here, we are asked to find the number of revolutions. Because 1 rev=2π rad, we can find the number of revolutions by finding θ in radians. We are given α and t, and we know ω_{0} is zero, so that θ can be obtained usingSolution for (d)
The number of meters of fishing line is x, which can be obtained through its relationship with θ:Discussion
This example illustrates that relationships among rotational quantities are highly analogous to those among linear quantities. We also see in this example how linear and rotational quantities are connected. The answers to the questions are realistic. After unwinding for two seconds, the reel is found to spin at 220 rad/s, which is 2100 rpm. (No wonder reels sometimes make highpitched sounds.) The amount of fishing line played out is 9.90 m, about right for when the big fish bites.Example 2. Calculating the Duration When the Fishing Reel Slows Down and Stops
Strategy
We are asked to find the time t for the reel to come to a stop. The initial and final conditions are different from those in the previous problem, which involved the same fishing reel. Now we see that the initial angular velocity is ω_{0 }= 220 rad/s and the final angular velocity ω is zero. The angular acceleration is given to be α = 300 rad/s^{2}. Examining the available equations, we see all quantities but t are known in ω = ω_{0}+αt, making it easiest to use this equation.Solution
The equation statesω = ω_{0 }+ αt.
We solve the equation algebraically for t, and then substitute the known values as usual, yieldingDiscussion
Note that care must be taken with the signs that indicate the directions of various quantities. Also, note that the time to stop the reel is fairly small because the acceleration is rather large. Fishing lines sometimes snap because of the accelerations involved, and fishermen often let the fish swim for a while before applying brakes on the reel. A tired fish will be slower, requiring a smaller acceleration.Example 3. Calculating the Slow Acceleration of Trains and Their Wheels
Strategy
In part (a), we are asked to find x, and in (b) we are asked to find ω and v. We are given the number of revolutions θ, the radius of the wheels r, and the angular acceleration α.Solution for (a)
The distance x is very easily found from the relationship between distance and rotation angle:x = rθ.
x = rθ = (0.350 m)(1257 rad) = 440 m
Solution for (b)
We cannot use any equation that incorporates t to find ω, because the equation would have at least two unknown values. The equationDiscussion
The distance traveled is fairly large and the final velocity is fairly slow (just under 32 km/h).Example 4. Calculating the Distance Traveled by a Fly on the Edge of a Microwave Oven Plate
Strategy
First, find the total number of revolutions θ, and then the linear distance x traveled.Solution
Entering known values intoDiscussion
Quite a trip (if it survives)! Note that this distance is the total distance traveled by the fly. Displacement is actually zero for complete revolutions because they bring the fly back to its original position. The distinction between total distance traveled and displacement was first noted in OneDimensional Kinematics.Check Your Understanding
Solution
Rotational kinematics (just like linear kinematics) is descriptive and does not represent laws of nature. With kinematics, we can describe many things to great precision but kinematics does not consider causes. For example, a large angular acceleration describes a very rapid change in angular velocity without any consideration of its cause.Section Summary
 Kinematics is the description of motion.
 The kinematics of rotational motion describes the relationships among rotation angle, angular velocity, angular acceleration, and time.
 Starting with the four kinematic equations we developed in the OneDimensional Kinematics, we can derive the four rotational kinematic equations (presented together with their translational counterparts) seen in Table 1.
 In these equations, the subscript 0 denotes initial value (${x}_{0}\\$and${t}_{0}\\$are initial values), and the average angular velocity$\bar{\omega}\\$and average velocity$\bar{v}\\$are defined as follows:
$\bar{\omega }=\frac{{\omega }_{0}+\omega }{2}\text{ and }\bar{v}=\frac{{v}_{0}+v}{2}\\$.
Problems & Exercises
2. Suppose a piece of dust finds itself on a CD. If the spin rate of the CD is 500 rpm, and the piece of dust is 4.3 cm from the center, what is the total distance traveled by the dust in 3 minutes? (Ignore accelerations due to getting the CD rotating.)
3. A gyroscope slows from an initial rate of 32.0 rad/s at a rate of 0.700 rad/s^{2}. (a) How long does it take to come to rest? (b) How many revolutions does it make before stopping?
4. During a very quick stop, a car decelerates at 700 m/s^{2}.
(a) What is the angular acceleration of its 0.280mradius tires, assuming they do not slip on the pavement?
(b) How many revolutions do the tires make before coming to rest, given their initial angular velocity is 95.0 rad/s?
(c) How long does the car take to stop completely?
(d) What distance does the car travel in this time?
(e) What was the car’s initial velocity?
(f) Do the values obtained seem reasonable, considering that this stop happens very quickly?
5. Everyday application: Suppose a yoyo has a center shaft that has a 0.250 cm radius and that its string is being pulled.
(a) If the string is stationary and the yoyo accelerates away from it at a rate of 1.50 m/s^{2}, what is the angular acceleration of the yoyo?
(b) What is the angular velocity after 0.750 s if it starts from rest?
(c) The outside radius of the yoyo is 3.50 cm. What is the tangential acceleration of a point on its edge?
Glossary
 kinematics of rotational motion:
 describes the relationships among rotation angle, angular velocity, angular acceleration, and time
Selected Solutions to Problems & Exercises
1. (a)3. (a) 45.7 s (b) 116 rev
5. (a) 600 rad/s^{2 }(b) 450 rad/s (c) 21.0 m/s