Download Newton`s Laws

Rotational (Angular) Motion
Linear Motion vs. Rotational Motion
Linear (Translational) Motion – involves an
object moving in a straight line
Rotation (Angular) Motion - involves an
object rotating about an axis
(examples: merry-go-round, rotating Earth, a
spinning skater, a top, a turning wheel)
Angular Quantities
Where,
1 rad
r = radius
There
is (m)
6.28
S =arc length (m)
radians
or 2π
Θ = radian (unit less)
radians in 1 circle
s
θ
r
0.28
axis
What is a radian?
This is also
A projection of 1 radius (or
equivalent
to 360°
amount of angular
displacement it takes to go)
around a circle
This equation relates linear and angular quantities!
Arc Length
(m)
S = θr
Angular Displacement
(radians)
Radius (m)
Relate back to something you know!
What do you call the
arc length of the entire
circle?
θ
1 complete circle = 360° = 2 π radians
r
S = θr
S = 2π r
The circumference is
basically the arc
length of the entire
circle
C = 2π r
The Radian!
Is radian really a unit?
NOPE! We only use radians to keep our “unit of
measurement” in mind.
(a.k.a make sure we are not in degrees)
It is SUPPPER
important you know
how to change your
calculator to and from
radians - degrees
Converting Quantities
You need to know how to convert!
1 radian = ___ degrees?
180°= ___ radians?
4 revolutions = ___ radians?
key concepts in rotational motion :
its JUST like Linear motion!
You can relate the two!
Don’t let symbols intimidate you!
Linear and Rotational Displacement
(s and “x” or “d” have the same idea)
Linear
Rotational
x
x
Linear and Rotational Displacement
(s and “x” have the same idea)
Linear
S =θ r
Arc Length
(m)
Up, N, E, right = +
Down, S, W, left = -
Rotational
θ = S/r
Angular Displacement
(radians)
CCW = +
CW = -
Linear and Rotational Velocity
Radians per second (rad/s)
Linear
Rotational
x
x
v=
x = Δx = Δs
t
t
t
ω=
= Δθ
t
Linear and Rotational Velocity
Linear
Rotational
v = x = Δx = Δs
t
Δt Δt
ω = θ = Δθ
t
Δt
Measured in m/s
Up, N, E, right = +
Down, S, W, left = -
v
Measured in rad/s
CCW = +
CW = -
ω
Linear and Rotational Acceleration
Linear
Rotational
Linear and Rotational Acceleration
Linear
a = Δv = vf -vi
Δt Δt
Measured in m/s2
Up, N, E, right = +
Down, S, W, left = -
Rotational
α
= Δω = ωf-ωi
Δt
Δt
Measured in rad/s2
CCW = +
CW = -
Δs
What do you know about
the linear displacement?
Angular?
Δs
Δs
θ
A
B
Objects
C
What do you know about
the linear velocity?
Angular?
v = x = Δx = Δs
t
Δt Δt
What do you know about
the linear acceleration?
Angular?
a = Δv = vf -vi
Δt Δt
α
ω = θ = Δθ
t
Δt
= Δω = ωf-ωi
Δt
Δt
There is a parallel between linear and
rotational motion. Most of the
equations will look “Familiar”,
however, we will change linear values
into angular values.
The math is done the same way
Rotational Kinematic Translations
Every term in a given linear equation has a
corresponding term in the rotational
equation
Angular Motion Graphs
(resemble the motion graphs we saw during the kinematics unit)
Angular Motion Graphs
θ
Δθ =ω
t
The SLOPE of a Angular
Displacement vs Time graph is
the Angular Velocity!
Time
Δω = α
t
ω
Time
The SLOPE of a Angular Velocity
vs Time graph is the Angular
acceleration!
The Area under the line is the
Angular Displacement!
Area =ω t =Δθ
Relationships between Linear and Angular
Quantities
Displacement
S=θr
Velocity
v = ωr
T
Acceleration
a = αr
T
Every point on a rotating object
has the same angular motion
Every point on a rotating object
does not have the same linear
motion
For any circle you can convert to and from angular
and linear velocity
v = ωr
ω = v/r
We can apply the same principles to acceleration
a = αr
α = a/r
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Rotational Dynamics
In linear motion, and
object requires a force to
accelerate.
In rotational motion, a
torque is required
Forces cause accelerations
Torques cause angular accelerations
Force and torque are related
t
Torque
Torque, (greek letter “tau”), is the tendency of
a force to rotate an object about some axis
t=rF
t = r F (sinθ)
For Forces at angles
τ
r
F
Sin(θ)
is torque
is the length from the axis of rotation
is the force
is the angle between F and r
Sign Convention for Torque
Using the Right hand Rule!!!
Positive torque:
Counter-clockwise,
out of page
ccw
cw
Negative torque:
clockwise, into page
Things to know about Torque
Torque is a Vector! It can be + or –
Measured in Newton-meters (N.m)
If the Force is at an angle, only the
component PERPENDICULAR to the lever
arm is applied.
If more an one torque is applied, just add
τ
them as vectors. If net=0 , then there is
no angular acceleration.
Torque is determined by Three Factors:
* The magnitude of the applied force.
* The angle of the applied force.
* The location of the applied force.
The forces nearer the
end of the wrench
have greater torques.
Location of force
20 N
20 N
20 N
How we see Torque
Torque – and the lever arm
The lever arm, d, is the perpendicular distance from the
axis of rotation to a line drawn along the direction of the
force
d = r sin q
This also gives t = r F sin q
τ net is Στ so…. Στ = τ1 + τ2 + ….etc.
We can break it down further
Στ = F1r1(sinθ) + F2r2(sinθ) +…etc
2
+
-
+
1
3
Στ = 30(2)(sin45°) + 25(0)(sin30°) - 10(2)(sin20°)
A boy of mass 45 kg and his dog of mass 25 kg stand on a diving
board. If the boy stands 2.0 meters from the pivot of the board,
and the dog stands 1.2 meters from the pivot, what is the net
torque on the board?
Στ = τdog +τboy
2m
τ = Fr
1.2m
Στ = (-441*2)+(-245*1.2)
245 N
441 N
Στ = -1176 N*m
Come in.. Get both papers on the
back of table…
we got notes and a virtual lab today
Rotational Equilibrium is a balance.
When an object is in equilibrium the object is not
accelerating.
Two bubbles are on a seesaw. How far from the center does the
yellow bubble have to sit to balance out the seesaw?
Force of each.
W = mg
W = 294 N
W = 196 N
M=
30kg
Set Torque equal
on both ends.
Fr 1 = Fr 2
M=
(294)(3) = (196) (r2)
20kg
Solve for r2
?m
3m
882 = r 2
196
R2 = 4.5 meters
What do we know so far?
Angular
Quantities
S
Θ
r
VT
aT
ω
α
τ
Angular and Linear
Formulas
S=θr
θ = s/r
v = Δs
Δt
ω = Δθ
Δt
a =vf -vi
Δt
α = ωf-ωi
Δt
Angular and Linear
Conversions
v = ωr
T
Angular Kinematics
a = αr
T
Torque
t=rF
t = r F (sinθ)
Torque and Angular Acceleration
When a rigid object is subject to a net torque (Στ ≠ 0), it
undergoes an angular acceleration
The angular acceleration is directly proportional to the
net torque
The more Torque, the more angular acceleration
The relationship is similar to ∑ F = ma
Newton’s Second Law
Newton’s Second Law for a Rotating Object
Net
Torque
Angular
acceleration
Moment
of Inertia
Moment of Inertia??
An objects resistance to change its rotation…
similar to mass in linear motion.
Moment of Inertia
the moment of inertia depends on the quantity of matter and its
distribution in the rigid object and its distance from the axis of
rotation.
Moment
of Inertia
Distance from
axis
kg.m2
Mass
The units for Moment of Inertia is kg.m2
Moment of Inertia
Objects with different rotational
inertias move differently
L
m
m
2
m1r
m
What is the
moment of inertia
for this?
ΣΙ=
ΣΙ=
ΣΙ=
2
2
Σ Ι = ¼mL + ¼mL
2
Σ Ι = ½ mL
2
Σmr
2
2
+ m2r + m3r
2
2
2
m(½L) + m0 + m(½L)
L
m
m
m
ΣΙ=
ΣΙ=
2
2
Σ Ι = ¼mL + mL
2
Σ Ι = 5/4 mL
What is the
moment of inertia
for this?
ΣΙ=
2
Σmr
2
2
2
m1r + m2r + m3r
2
2
2
m0 + m(½L) + m(L)
Is the CENTER of MASS the same as
the MOMENT of INERTIA?
ΣΙ=
2
Σmr
NO!
X = m1x1 + m2x2 +….
(m1 + m2+….)
CM
The moment of inertia refers to the
mass distribution of an object whereas
the center of mass refers to the
location where the mass is
concentrated at.
Ok, back to the 2nd law.
Ex. Consider a solid cylinder of mass 3kg and radius of 2m.
When a Force of 20 N is applied, what is the angular
acceleration?
F
m
r
m = 3kg
r= 2m
F= 20 N
I = ½ mr2
F r = ½ mr2 α
- F = ½ mr α
α = - 2F
mr
α = - 2(20)
(3*2)
α = -6.67 rad/s2
If the cylinder from the previous example started from rest…
What is its angular displacement after 2 seconds?
ωo = 0 rad/s
t = 2s
α = -6.67 rad/s2
θ=?
θ = ωot + ½αt2
θ = (0*2) + ½ (-6.67 *22)
θ = -13.34 rad
τ
What is the acceleration of the block?
M
(This problem uses both rotational and
linear motion)
r
rotational
m
Linear
M
-T
r
m
+T
mg
How to solve:
1. Draw FBD for
all objects with
mass.
2. Assign
Direction
3. Relate α to
α =aT /r
4. Combine
equations and
solve.
mg - T= ma
T r = ½ Mr2 α
mg - ½ Ma = ma
T = ½ Mr (a)
r
mg= ma + ½ Ma
mg= a (m+ ½ M)
T = ½ Ma
mg
(m+ ½ M)
= a
Try this one…
What is the acceleration of m2
M1
r
M
M2
Rotational Kinetic Energy
Rotating objects can have both rotational and translational kinetic energies
Object will rotate through an
angular displacement
Both will
result in
kinetic
energy
Object will translate through
a linear displacement
KETOTAL = KE + KER
KETOTAL =
2
½mv
“Rotational”
+
2
½Iω
N
FF
What force causes Torque?
FRICTION!
FBD of an object rolling down a hill.
The object will “slip” when
going down the incline
resulting in a change in
angular velocity – and
producing a torque.
Does that mean rolling object move faster than
a translating object?
Object 1: Translating
h
No Friction
Which will travel
faster?
Object 2: Rotating
h
Friction
Both objects start out with the same potential energy =mgh.
But object 1 will be faster because all the potential energy is
converted to kinetic energy whereas in object 2 the total potential
energy is split between kinetic energy and rotational kinetic energy.
Object 1
Object 2
PE = KE
PE = KE + KER
mgh = ½mv2
mgh = ½mv2 + ½Iω2
v=
2𝑔ℎ
v=
4
𝑔ℎ
3
Angular Momentum
Linear Momentum
Angular Momentum
p = mv
L = Iω
τ = ΔL = Δmvr
Δt
Δt
Angular Momentum is conserved
unless a net external torque is
present. A torque causes a
change in momentum over time
If Στ = 0, then Li = Lf
If τ ≠ 0, then Li ≠ Lf