Rotational Kinematics and Rotational Kinematics and InertiaInertia

Circular MotionCircular Motion

Angular displacement = 2-1

How far it has rotatedUnits radians 2 = 1 revolution

Angular velocity = t How fast it is rotatingUnits radians/second 2 = 1 revolution

Angular acceleration is the change in angular velocity divided by the change in time. α = tHow much is it speeding up or slowing downUnits radians/second2

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Period, FrequencyPeriod, Frequency

FrequencyNumber of revolutions per sec

Period =1/frequency T = 1/f = 2Time to complete 1 revolution

Circular to LinearCircular to Linear(Why use Radians)(Why use Radians)

Displacement s = r in radians)

Speed |v| = s/t = r /t = rDirection of v is tangent to circle

cceleration |a| = rα

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Angular AccelerationAngular AccelerationIf the speed of a roller coaster car is 15 m/s at the top of a 20 m loop, and

25 m/s at the bottom. What is the cars average angular acceleration if it takes 1.6 seconds to go from the top to the bottom?

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Comparison to 1-D Comparison to 1-D kinematicskinematics

Angular Linear

constant

t0

0 021

2t t

constanta

v v at 0

x x v t at 0 021

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And for a point at a distance R from the rotation axis:

x = Rv = Ra = R

Example: cd playerExample: cd player The CD in your disk player spins at about 20 radians/second. If it

accelerates uniformly from rest with angular acceleration of 15 rad/s2, how many revolutions does the disk make before it is at the proper speed?

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Example: 48x cd-romExample: 48x cd-rom A 48x cd-rom spins at about 9600 rpm. If it takes 1.5 sec. to get

up to speed, what is the angular acceleration? How many revolutions does the disk make before it is at the proper speed?

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Rotational Inertia, Rotational Inertia, II Tells how difficult it is get object spinning. Just

like mass tells you how difficult it is to get object moving.Fnet= m a Linear Motionτnet = I α Rotational Motion

I = miri2 (units kg m2)

Note! Rotational Inertia depends on what you are spinning about (basically the ri in the equation).

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Inertia RodsInertia RodsTwo batons have equal mass and length. Which will be “easier” to spin

A) Mass on endsB) SameC) Mass in center

I = m r2 Further mass is from axis of rotation, greater moment of inertia (harder to spin)

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Example: baseball batExample: baseball bat

Rotational Inertia TableRotational Inertia Table For objects with finite number of

masses, use I = m r2. For “continuous” objects, use table below.

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