Download Rotational Kinematics

Kinematics vs. Rotational Motion
When discussing kinematics, what are some
common vector quantities we discuss?
• Displacement
Angular displacement
• Velocity
Angular velocity
• Acceleration
Angular acceleration
• Force
Torque
• Momentum
Angular momentum
ALL of these vectors can be described in terms of
rotational motion!
How do we describe angular
displacement?
Lets say I have a disk rotating clockwise…
From math class, how do we
describe the amount a circle
rotates?
• Radians (rad)
• Symbol: θ
r
Ex: The dot has an angular displacement of 90o
or π/2 rad or 1.57 rad
What exactly is 1 radian?
• One radian is the angle created by an arc
whose length is equal to the radius.
When l = r the angular displacement is one
radian.
Therefore,
l
q=
r
Make the following conversions
1 revolution to radians
60 degrees to radians
4.5 revolutions to radians
48 degrees to radians
2π rad
1.05 rad or π/3 rad
9π rad or 28.3 rad
0.84 rad
A falcon can distinguish objects that extend a
minimum angular displacement of 3 x 10-4
rad.
a. How many degrees is this?
b. How small an object can the bird
distinguish when flying a height of 100 m?
a.
o
360
(3´10-4 rad)´
=
2p rad
b.
l =100m´ (3´10-4 rad) l = 3´10-2 m
0.017o
How do we describe “angular velocity”?
Thinking about velocity for a second, how do we
determine velocity?
• Displacement/time = velocity
So it should be intuitive that angular velocity is
how many radians are covered in a certain
period of time.
q Symbol: ω (omega)
w=
Units: rads/sec
t
How is angular velocity related to
linear velocity?
w=
q
t
l
v=
t
Can we express both in one equation?
• Yes, we can combine equations and simplify
using our expression for radians!
• Related:
v = rw
l
q=
r
l = rq
Angular Acceleration
How did we determine angular velocity?
We simply applied the linear velocity equation in
terms of radians! So what is angular acceleration?
w
- wi
D
w
f
Symbol: α (alpha) a =
=
t
t
Units: rad/sec2
Look familiar?
Tying together relationships
What expression related angular displacement to
displacement and angular velocity to velocity?
l
q = ® l = rq
r
v = rw
So, what do you think is
the equation that relates
angular and linear
acceleration?
How is each relationship similar?
a = ra
How do all the kinematic equations
relate to rotational motion?
Kinematics
Rotational
q
d
v 
t
v 2f  vi2  2ad
t
2
2
w f = wi + 2aq
v f  vi  at
w f = wi + a t
v
vi  v f
2
1 2
d  vi t  at
2
w=
w=
wi + w f
2
1 2
q = wi t + a t
2
Making things even easier!
rev 2p rad
rad
A lot of the time, rotational
5.5
´
= 34.56
sec 1rev
sec
questions give you the
angular velocity as “rpm” or • “rps” is simply frequency,
“rps”.
so this conversion can be
written as: w = 2p f
We can convert this since
one rotation is 2π radians!
What is 5.5 rps in rad/s?
Example
A centrifuge rotor is accelerated from rest to
20,000 rpm in 30 s.
a. What is the average angular acceleration?
a. Through how many revolutions has the centrifuge
rotor turned during its acceleration period,
assuming constant angular acceleration?
Solutions
a. ωi = 0, so that leaves just ωf
rev 1min
rev
20, 000
´
= 333.3
min 60sec
sec
ωf=2πf = 2π rad/rev x 333.3 rev/s
ωf ≈ 2100 rad/s
w f = wi + a t
α = 70 rad/sec2
b. Recall one revolution is 2π
radians.
1 2
q = wi t + a t
2
θ = 0 + .5(70 rad/s2)(30s)2
θ = 3.15 x 104 rad
1rev
3.15´10 rad ´
= 5000 rev
2p rad
4
Examples
Summary
1. How is linear acceleration, velocity and
displacement related to it’s rotational counterpart?
2. How is frequency related to angular velocity?
3. What is “rps” and “rpm” stand for? What do both
represent?
How do we describe Newton’s
second law for rotation?
What is Newton’s Second Law?
• F = ma
So, what did we say is the
rotational equivalent to force?
• TORQUE!!!
Weislearned
about
torque
earlier,
but
This
the exact
same
equation
that
wehow can we
describe
it in terms
of rotational
motion?
How did
learned
before,
but now
we see how
it
we writeallthe
versionsones!
for ALL the
matches
therotational
other rotational
kinematic equations???
t = rF
Deriving a rotational expression for
torque
So now we have “τ = rF”, but if torque
represents the 2nd law for rotation, then what
about mass?
How can we write force in terms of rotation?
• F = ma
Now we can substitute into
• F = m x rα
the equation for torque!
• τ = mr2α
Rotational Inertia
τ = mr2 x α
So, lets compare this to F = ma
• Torque is rotational force
• Angular acceleration is rotational acceleration
• (mr2) is rotational inertia with symbol “I”
I = å mr
2
t = rF ® t = mr a ® å t = I a
2
Demo: Rotational Inertia
Which has more rotational inertia “I”?
• Rotational motion
measures how hard it is
to change angular
velocity.
• It’s based on mass and
it’s distribution regarding
the axis of rotation.
The cylinder is faster, so it must have less
rotational inertia. It was easier to move!
Two weights on a bar: different axis,
different “I”
Two weights of mass 5 kg
and 7 kg are mounted 4 m
apart on a light rod
(whose mass can be
ignored). Calculate the
moment of inertia when
rotated about an axis
2
I
=
Σmr
halfway between the
I = (5kg)(2m)2 + (7kg)(2m)2
weights.
I = 48.0 kg.m2
Different Axis
Calculate the moment of
inertia now when rotated
about an axis 0.5 m to the
left of the 5 kg mass.
How does the inertia
I = Σmr2
I = (5kg)(.5m)2 + (7kg)(4.5m)2 added by the mass
close
to
the
axis
.
2
I = 143 kg m
compare to the mass
farther away?
Summary
1. What is Newton’s second law for rotational motion?
2. How can we define rotational inertia?
3. What affects the rotational inertia “I”?
4. What was the difference between parts “a” and “b”
in the practice questions?