Transcript Projectile Motion

Rotational Motion
Rotational motion is analyzed in term of two properties:
• axis of rotation
• rotational velocity
Rotational Velocity
Rotational velocity can be measured as:
• period of rotation – time for one rotation
• frequency of rotation – rotations in one second
T = period = 1 / frequency
f - frequency = 1 / period
Example: period T = 0.25 sec/rot, f = 4 rot/sec
Frequency and Period
T = period = 1 / frequency
f - frequency = 1 / period
Example: A CD player spins at a rate of 12 r/s.
What is its period and frequency?
12 r/s is a measure of frequency
Period T = 1 / 12 r/s = 0.083 s
Units of Frequency
Heinrich Hertz provided much of our knowledge
of waves. We honor his work by naming the unit
frequency the Hertz (Hz)
In the previous problem, the CD is spinning at a
rate of 12 Hz.
FM radio stations use MHz for their broadcast
frequency. (i.e. WPGC 95.5 MHz)
Angular Speed
Angular speed is the angle through which a
rotating object spins over time.
We use Greek letters to identify the variables.
Angular speed – ω (omega)
Rotation angle –  (theta)
ω=/t
Angular Acceleration
Angular velocity and acceleration use
equations that are similar to linear velocity
and acceleration.
Linear
v=d/t
vf = vi + a t
d = vi t + ½ a t2
vf 2 = v i 2 + 2 a d
Angular
ω=/t
ωf = ωi +  t
 = ωi t + ½  t2
ωf2 = ωi2 + 2 
Torque
Torque is a force that is applied to a
rotating system that causes it to turn.
When we studied forces, we began
by drawing a force diagram.
Tangential Force
Torque is a tangential force applied at
perpendicular to the axis of rotation.
Tangential force
Lever arm
Torque Equation
Torque is the product of the
tangential force and the
perpendicular distance to the
axis of rotation.
τ = F r sin 
Torque Equilibrium
A St. Bernard and a Chihuahua balance
on a plank as shown
Torque Equilibrium
These forces don’t intersect
Each force, acting alone, would cause
the seesaw to ROTATE around the
pivot
Torque Equilibrium
Their Lever
Arms
St. Bernard
Force
Chihuahua
Force
If the bar is to balance…
…the torques must cancel
Fr=Fr
Rotational Inertia
Moving objects have linear inertia,
spinning objects have rotational inertia.
Rotational inertia is the product of
mass and radius squared.
I = k m r2
Rotational Inertia
Different shapes have a different constant:
The greater the constant, the more inertia.
Rod – k = 1/12
Disc – k = ½
Ring – k = 1
Angular Momentum
Moving objects have linear momentum,
spinning objects have angular momentum.
L=Iω
Angular momentum is the product of
rotational inertia and angular velocity.
Conservation of Momentum
The momentum is conserved, so a decrease
in the radius and rotational inertia causes
an increase in angular velocity.
Iω
I
ω
You see this in spinning figure skaters and gymnasts.
Right Hand Rule
If you curve the fingers of your right hand in
the direction of rotation, your thumb points
in the direction of angular momentum.