Chapter 10 – Rotational Kinematics & Energy

10.1 – Angular Position (θ)

• In linear (or translational) kinematics we looked at the position of an object (Δx, Δy, Δd…)

• We started at a reference point position (xi) and our definition of position relied on how far away from that position we are.

• Likewise, our angular position relies on how far we’ve rotated (Δθ) from a reference line.

10.1 - Angular Position (θ)

10.1 – Angular Position (θ)

Degrees and revolutions:

10.1 – Angular Position (θ)

Arc length s, measured in radians:

Arc length is how far (length) we’ve moved around the circle (arc).

10.1 – Angular Velocity (ω)

• Change in linear position of an objet over time is velocity. – How quickly we

change position.

Linear Velocity Rotational Velocity• Change in angular

position of an object over time is angular velocity.– How quickly angle

changes.

10.1 – Angular Velocity (ω)

Sign Convention:

10.1 – Angular Velocity (ω)

A drill bit in a hand drill is turning at 1200 revolutions per minute (1200 rpm). Express this angular speed in radians per second (rad/s)

A) 2.1 rad/sB) 19 rad/sC) 125 rad/sD) 39 rad/sE) 0.67 rad/s

Question 10.1a Bonnie and Klyde I

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BonnieKlyde

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 complete revolution every2 seconds.Klyde’s angular velocity is:

a) same as Bonnie’s

b) twice Bonnie’s

c) half of Bonnie’s

d) one-quarter of Bonnie’s

e) four times Bonnie’s

10.1 – Angular Velocity (ω)

10.1 – Angular Acceleration (α)

Linear Acceleration• Defined as how quickly our

velocity is changing per unit time.– When we speed up or slow

down.

Angular Acceleration• Defined as how our angular

velocity (ω) changes per unit time.– How fast we rotate, does that

speed up or slow down?– Ex: airplane propellers

• Really, really, REALLY dumb idea…

10.1 – Angular Acceleration (α)

10.1 – Angular Acceleration (α)

Sign Convention:

10.2 – Rotational KinematicsAnalogies between linear and rotational kinematics:

Example 10.2 (pg. 304)

If the wheel is given an initial angular speed of 3.40 rad/s and rotates through 1.25 revolutions and comes to rest on the BANKRUPT space, what is the angular acceleration of the wheel (assuming it’s constant)?

10.3 – Tangential Speed

What is tangential speed?Imagine riding a merry-go-round, and suddenly letting go before the ride stops. With what velocity will you fly off the merry-go-round?

Question 10.1b Bonnie and Klyde II

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BonnieKlyde

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?

10.3 – Centripetal Acceleration of Rotating Object

10.3 – Tangential Acceleration

10.3 – Tangential & Centripetal Acceleration

This merry-go-round has BOTH tangential and centripetal acceleration.

10.1 – 10.3 SummaryArch Length

Average Angular Velocity

Instantaneous Angular Velocity

Period of Rotation

Average Angular Acceleration

Instantaneous Angular Acceleration

10.1 – 10.3 SummaryLinear Kinematics

(a = constant)Rotational Kinematics

(α = constant)

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

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

10.5 – Rotational Kinetic Energy

Linear Kinetic Energy• Depends on an objects

linear speed.

• NOT valid for a rotating object because v is different for points of various distances from the axis of rotation.

Rotational Kinetic Energy• Depends on an objects

angular speed.

10.5 – Moment of Inertia

• Rotational Kinetic Energy depends on ω2 and r2. AKA the distribution of mass of the rotating object.

• Moment of Inertia (I) – • Rotational Kinetic Energy can be rewritten as

10.5 – Moment of Inertia

• Moment of Inertia is the distribution of mass throughout the rotating object.

10.5 – Moment of InertiaCalculate the Moment of Inertia of this object.

Conceptual Checkpoint 10-2

10.5 – Moment of Inertia of Various ObjectsMoments of inertia of various regular objects can be calculated (pg. 314):M = total massR = radiusL = Length

10.5 – Kinetic Energy ComparisonKinetic Energy Linear Quantity Angular Quantity

Speed Variable v ω

Mass Variable m I

Final Equation ½ mv2 ½Iω2

10.6 – Conservation of EnergyThe total kinetic energy of a rolling object is the sum of its linear and rotational kinetic energies:

Example 10.5 (pg 316)

What’s the total Kinetic Energy of this 1.20 kg rolling object?

What’s the speed of this object when it reaches the bottom of the ramp?