Rotational Kinematics

•  See intro in previous lecture (online at course website under the “Instructor” tab).

Direction of rotation

Clockwise:

Counterclockwise:

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Δθ < 0

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Δθ > 0

S.I. units : radians (rad)

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θ in rad( ) =Arc length

Radius=sr

Sign Convention:

à s = r✓

C&J conceptual question 2 A meter stick is rotated about the end labeled 0.00 cm, so that the other end of the stick moves through an arc length of 7.33 cm. Through what arc length s does the 25.0-cm mark on the stick move?

a)  7.33 cm

b)  1.83 cm

c)  29.3 cm

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Angular velocity: How fast is the rigid body rotating?

S.I. units of angular velocity: rad/s

Instantaneous angular velocity:

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ω = limΔt→0

ΔθΔt

Angular velocity: How fast is the rigid body rotating?

S.I. units of angular velocity: rad/s

Clockwise: Counterclockwise:

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ω < 0

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ω > 0

8.7 The Vector Nature of Angular Variables

Right-Hand Rule: Grasp the axis of rotation with your right hand, so that your fingers circle the axis in the same sense as the rotation. Your extended thumb points along the axis in the direction of the angular velocity.

Relation between angular velocity and period:

•  Recall: Period T is time to complete one revolution.

•  One revolution: θ = 2π rad

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ω =2πT

C&J 8.11 The sun appears to move across the sky, because the earth spins on its axis. To a person standing on the earth, the sun subtends an angle of 9.28×10-3 rad. How much time (in seconds) does it take for the sun to move a distance equal to its own diameter?

C&J 8.13 Two people start at the same place and walk around a circular lake in opposite directions. One walks with an angular speed of 1.7x10-3 rad/s, while the other has an angular speed of 3.4x10-3 rad/s. How long will it be before they meet?

Angular acceleration S.I. unit: rad/s2

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α = limΔt→0

ΔωΔt

Instantaneous angular acceleration

sign of α

•  If the angular motion is speeding up, α and ω have the same sign.

•  If the angular motion is slowing down, α and ω have the opposite sign.

CYU #4 An electric clock is hanging on the wall in the living room. The battery is removed and the second hand comes to a halt over a brief period of time. Which one of the following statements correctly describes the angular velocity ω and the angular acceleration α of the second hand as it slows down?

α)  ω and α are both negative. β)  ω is positive and α is negative. χ)  ω is negative and α is positive. δ)  ω and α are both positive.

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Equations of kinematics for constant acceleration

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v = v0 + at

x = x0 + 12 v0 + v( )t

x = x0 + v0t + 12 at2

v2 = v02 + 2a x − x0( )

displacement: Δx = x - x0 initial velocity: v0 final velocity: v (constant) acceleration: a time: t

Now replace:

•  displacement with angular displacement

•  velocity with angular velocity

•  constant acceleration with constant angular acceleration

•  and leave time alone!

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Δx = x − x0 ⇒ Δθ =θ −θ0

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v0 ⇒ω0 v⇒ω

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a⇒α

The equations of rotational kinematics for constant angular acceleration

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ω = ω0 +αtθ = θ0 + 1

2 ω0 +ω( )t

θ = θ0 +ω0t + 12αt

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ω2 = ω02 + 2α θ−θ0( )

Problems -- reasoning strategy:

•  Make a drawing!

•  Note the positive and negative directions.

•  Write down the values that are given for any of the 5 variables, with the correct sign.

•  You need values for at least 3 of the variables. Choose the appropriate equation.

•  Remember that there may be two possible answers.

C&J 8.23 A wind turbine is initially spinning at a constant angular speed. As the wind's strength gradually increases, the turbine experiences a constant angular acceleration of 0.140 rad/s2. After making 2870 revolutions, its angular speed is 137 rad/s. (a) What is the initial angular velocity of the turbine? (b) How much time elapses while the turbine is speeding up?

A wheel which is initially at rest starts to turn with a constant angular acceleration. After 4 seconds it has made 4 complete revolutions. How many revolutions has it made after 8 seconds?

A.  8 B.  12 C.  16

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Homework

•  read 8.2, 8.3, 8.4 and 9.1, 9.2

•  C&J 8.22

Note: Quick questions have long answers. Therefore please try to see me during office hours at 2:30-4:30pm Tuesdays in Allen 514 or make an appointment.