Transcript Rotational Motion

Rotational Motion
WHS
Lee Wignall
Important Terms and Concepts
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Radians/degrees: how to convert from one to another
Rotational motion: what is it?
Arc length
Angular displacement (in radians)
Angular velocity (ω: omega)
Angular acceleration (α: alpha)
Rotational motion equations vs. linear motion equations
Tangential velocity/acceleration
Centripetal acceleration
Causes of rotational motion (gravity/torque)
Conservation of Angular Momentum
Basic Quantities
So far we’ve done everything in degrees, but
now we will make a change to radians for
rotational motion.
Why? Because things usually “turn” more
than just one revolution (360 degrees).
Converting from degrees to radians:
2π
360
=
θr
Θd
so,
θr = θd π
180
Arc Length
Arc length (m)
Radius (m)
angle (radians)
Angular Velocity
Linear equation for velocity:
change in distance
change in time
Angular velocity:
change in angle
change in time
“omega”: radians/s
“angular displacement”
ω = Δθ
Δt
Angular Velocity
Angular velocity:
“angular displacement”
change in angle
change in time
“omega”
ω = Δθ
Δt
COUNTER CLOCKWISE
=
positive direction (+)
CLOCKWISE
=
negative direction (-)
Angular Acceleration
Linear equation for acceleration:
change in velocity
change in time
Angular acceleration:
change in angular velocity
change in time
“alpha”: radians/s2
α = Δω
Δt
Angular vs. Linear Motion Equations
Sample Rotational Motion Problem #1:
A wheel is slowly turning at 1.0 rad/s when an outside force
causes the angular velocity to increase to 4.0 rad/s. If it takes
5 full revolutions of the wheel to reach it’s final angular
velocity, what was the angular acceleration provided by the
force?
Sample Rotational Motion Problem #2:
A helicopter blade accelerates at 35.5 rad/s2. If the initial
angular velocity of the blade was 5 rad/s, how long will it take
the blades to complete 1500 turns?
• What will be its angular velocity at that point?
• If it takes an angular velocity of 2000 rad/s to achieve
liftoff, how many seconds will that take?
Homework
Angular Displacement: page 247 (1-4)
Angular Speed:
page 248 (1-4)
Angular Acceleration:
page 250 (1-3)
Angular Kinematics:
page 252 (1-5)
Tangential Velocity and Acceleration
Radius (m)
vt = rω
Angular velocity (rad/s)
Tangential velocity (m/s)
at = rα
Tangential acceleration
(m/s2)
Angular acceleration (rad/s2)
Homework
Finish
Angular Displacement:
page 247 (1-4)
Angular Speed:
page 248 (1-4)
Angular Acceleration:
page 250 (1-3)
Angular Kinematics:
page 252 (1-5)
Today’s Work:
Tangential speed
page 255 (1-4)
Tangential acceleration:
page 256 (1-3)
Advanced Rotational Motion Problems
A mysterious planet with radius of 3.6 x 109 m has a gravitational pull
of 15 m/s2 but it’s spinning so fast that a 50 kg. rock at the equator
no longer falls downward. At what latitudes (north and south) will
the rock have a downward acceleration of 10 m/s2?
A 200 gram ball on a 0.75 meter string has an angular velocity of 1.5
rad/s. The person spinning it supplies a force that provides an
acceleration of 2.0 rad/s2. The string will break when the tension
reaches 150 N. How many seconds until the string breaks? At what
angle will the ball go flying off at? Assume you the ball begins at 0
degrees and spins counter-clockwise.
Conservation of Angular Momentum
The velocity is the TANGENTIAL
VELOCITY
So, the unit for angular momentum L
is:
“In any closed system, the total amount of angular momentum is
conserved.”
Formation of Galaxies
Horsehead Nebula
Formation of Galaxies
Crab Nebula
Formation of Galaxies
Formation of Galaxies
The Earth and tossed pizza dough do the same thing.
Figure Skaters
I is the “moment of inertia” which is also
called the radial mass. It’s a measure of
how much mass is how far away from the
axis of rotation.
Figure Skaters
Instead of thinking about moments of
inertia and angular velocity, we can think
about this spinning skater in terms of
radius and tangential velocity. It’s easier.
Figure Skaters
So, in order to conserve angular momentum L, if we DECREASE the
radial distance for the overall mass, then we will experience an
INCREASE in the tangential velocity, meaning we spin faster.