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1

Chapter 9

Rotation of Rigid Bodies

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Part 1

Rotational Motion

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3

Rotational kinematics

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A bird’s eyeA particular bird’s eye can distinguish objects no smaller than 3*10-4 rad.a) how many degrees is this?b) How small object can the bird see when flying as high as 100 m?

a) 3*10-4 * 57.3 = 0.0170

b) s = r*θ = 100 m *3*10-4 = 0.03 m

exampleθrs =

rev159.057.3rad1rad2rev1

3.572

360

0

00

deg

===

⋅≈=

π

θθπ

θ radrad

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5

Angular velocity

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Vector nature of angular velocity

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7

Linear and angular motion

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Hard DriveThe platter of a hard drive of a computer rotates at 7200 rpm (rpm = rev/min). a) What is the angular velocity (rad/s) of the platter?b) If the reading head is 3.0 cm from the rotational axis, what is the linear speed of the point on the platter below the reading head?c) If a single bit requires 0.5 μm of length along the direction of motion, how many bits per second can the writing head write when it is 3.0 cm from the axis?

example

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9

Hard Drive

a) f = (7200 rev/min)/(60 s/min) = 120 rev/s = 120 Hzthen ω = 2πf = 754 rad/s

b) linear speed v = rω = (0.03 cm)*(754 rad/s) = 22.6 m/sc) for each bit s = 0.5*10-6 m

at speed 22.6 m/s(22.6 m/s)/(0.5*10-6 m/bit) = 45*106 bit/s(or about 5.36 MB/s)

example

πω

ωππ

21,22 ====T

fvrT

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Rotation with constant acceleration

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11

Constant angular accelerationA rotor of a helicopter is accelerated from rest to 2000 rpm in 30 s.a) What is the average angular acceleration?b) Through how many revolutions has the rotor turned during its acceleration period (assuming constant acceleration)?a) ω = 2πf =

= (2π rad/rev)*(2000 rev/min)/(60 s/min) = 210 rad/sthen α = (210 rad/s)/(30 s) = 7 rad/s2. (about 1.1 rev/s)

b) θ = (7 rad/s2)*(30 s)2/2 = 3.15*103 rad = 500 rev.

example

f⋅= πω 2 tΔΔ= ωα2/2

00 tt αωθθ ++=

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Part 2

Rotational Kinetic Energy

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13

Rotational kinetic energy

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Moment of inertia

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15

Examples

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Test Device Inc.In 1975, Test Device Inc. was spin testing a sample of a solid steel rotor (a disk) of mass M = 272 kg, and radius R=0.38 m. When the sample reached an angular speed of 14000 rpm, the test system crashed (exploded) (a door to the test room was found on a nearby parking lot).How much energy was released in the explosion?angular speed: ω = 2πf = (2π rad/rev)*(14000 rev/min) / (60 s/min) = 1466 rad/sIdisk = 0.5MR2 = 19.64 kg/m2.K = 0.5*19.64 kg/m2 * (1466 rad/s)2 = 2.1*107 J.

example2

21 ωIK =

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17

Parallel Axis Theorem

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Moment of Inertia Calculation

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Uses of parallel axis theorem

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Example with Conservation of Energy

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21

A log on a roof after a hurricaneA log (like solid cylinder) of radius 10 cm and mass 12 kg starts from rest and rolls without slipping a distance of 6.0 m down a house roof that is inclined at 300. The rotational inertia for the cylinder is I=(1/2)mR2.

example

(a) What fraction of the total kinetic energy of the cylinder is due to rotation of the cylinder about its center as it leaves the house roof

(b) What is the speed of the ball at the end of the roof

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A log on a roof after a hurricane (cont.)example

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41

43

34

43

41

21

21

21

21

21

21

21

22

2222

22

2

22

===

=

=+=⎟⎠⎞

⎜⎝⎛+=

==

+=

tot

rotrottot K

KmvKmvK

ghv

mvmvmvRvmRmvmgh

RvmRI

Imvmgh

then

roof the of end the at ball the of speed

and using

energyofonconservati from

ω

ω