Rotational Kinematics

Angular Position

Degrees and revolutions:

Angular Positionθ > 0

θ < 0

Arc Length

Arc length s, from angle measured in radians: s = r θ

- Arc length for a full rotation (360o) of a radius=1m circle?

- What is the relationship between the circumference of a circle and its diameter?C / D = π

C = 2 π r

s = 2 π (1 m) = 2 π meters

1 complete revolution = 2 π radians

1 rad = 360o / (2π) = 57.3o

Why use radians?

• Doesn’t involve arbitrary choice of 360 degrees. There is another unit, the gon or gradian that is used in surveying:

• Radians useful for small angles:

5

90 100g

3 5

2 4

sin

c

3! 5!

o2 !

s 1! 4

Angular Velocity

Instantaneous Angular Velocity

Period = How long it takes to go 1 full revolution

Period T:

SI unit: second, s

Linear and Angular Velocity

Greater translation for same rotation

Bonnie and Klyde II

BonnieBonnieKlydeKlyde

a) Klyde

b) Bonnie

c) both the same

d) linear velocity is zero for both of them

Bonnie sits on the outer rim of amerry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one revolution every 2 seconds. Who has the larger linear (tangential) velocity?

a) Klyde

b) Bonnie

c) both the same

d) linear velocity is zero for both of them

Their linear speeds linear speeds vv will be

different because v = r v = r and

Bonnie is located farther outBonnie is located farther out

(larger radius r) than Klyde.

BonnieBonnie

KlydeKlyde

Bonnie and Klyde II

Bonnie sits on the outer rim of a merry-go-round, and Klyde sits midway between the center and the rim. The merry-go-round makes one revolution every 2 seconds. Who has the larger linear (tangential) velocity?

Angular Acceleration

Instantaneous Angular Acceleration

Rotational Kinematics, Constant Acceleration

If the acceleration is constant:

If the angular acceleration is constant:

v = v0 + at

Analogies between linear and rotational kinematics:

An object at rest begins to rotate with a constant angular acceleration. If this object rotates through an angle in the time t, through what angle did it rotate in the time t?

Angular Displacement IAngular Displacement I

½

a)

b)

c)

d) 2

e) 4

½

¼

¾

An object at rest begins to rotate with a constant angular acceleration. If this object rotates through an angle in the time t, through what angle did it rotate in the time t ?

a)

b)

c)

d) 2

e) 4

The angular displacement is = t 2 (starting from rest), and

there is a quadratic dependence on time. Therefore, in half the half the

timetime, the object has rotated through one-quarter the angleone-quarter the angle.

Angular Displacement IAngular Displacement I½

¼

½

¾

Which child experiences a greater acceleration?(assume constant angular speed)

Larger r: - larger v for same ω - larger ac for same ω

ac is required for circular motion.

An object may have at as well, which implies angular acceleration

Angular acceleration and total linear acceleration

Angular and linear acceleration

Rolling MotionIf a round object rolls without slipping, there is a fixed relationship between the translational and rotational speeds:

Rolling MotionWe may also consider rolling motion to be a combination of pure rotational and pure translational motion:

+ =

Jeff of the Jungle swings on a vine that is 7.20 m long. At the bottom of the swing, just before hitting the tree, Jeff’s linear speed is 8.50 m/s. (a) Find Jeff’s angular speed at this time.(b) What centripetal acceleration does Jeff experience at the bottom of his swing?(c) What exerts the force that is responsible for Jeff’s centripetal acceleration?

a)

b)

c) This is the force that is responsible for keeping Jeff in circular motion: the vine.