Chapter 10 – Rotational Kinematics & Energy
-
Upload
xander-whitaker -
Category
Documents
-
view
56 -
download
2
description
Transcript of 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)
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.
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
w
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 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…
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
w
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 – 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
10.4 - 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 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: