Transcript Lecture 8: Forces & The Laws of Motion

Lecture 18:
Angular Acceleration &
Angular Momentum
Questions of Yesterday
1) If an object is rotating at a constant angular speed which
statement is true?
a) the system is in equilibrium
b) the net force on the object is ZERO
c) the net torque on the object is ZERO
d) all of the above
2) Student 1 (mass = m) sits on the left end of massless seesaw
of length L and Student 2 (mass = 2m) sits at the right end.
Where must the pivot be placed so the system is in
equilibrium?
a) L/2
b) L/3 from the right (from Student 2)
c) L/3 from the left (from Student 1)
d) the system cant be in equilibrium
Torque & Angular Acceleration
Force causes
linear acceleration
TORQUE causes
angular acceleration
A system is in equilibrium (a = 0, a = 0)
when
∑F = 0 and ∑t = 0
F = ma
t = ?a
What is the induced angular acceleration a of an object due to
a certain torque t acting on an object?
t = rFsinq
aT = ra
Torque & Angular Acceleration
FT
FT = maT
r
m
FTr = mraT
t = r*Fsinq
t = mr2a
aT = ra
mr2 =
Moment of Inertia (I)
Moment of Inertia (I)
of a mass m rotating about an axis
at a distance r from the axis
= mr2
Moment of Inertia
t = mr2a
FT
t = t1 + t2…
t = m1r12a1 + m2r22a2…
m2 r2
m1
r1
a1 = a2 = a
t = (m1r12 + m2r22…)a = (mr2)a
I = m1r12 + m2r22… = mr2
t = Ia
Moment of Inertia
depends
on axis of rotation!!!
Moment of Inertia of a Rigid Body
What is the Moment of Inertia of a rotating rigid body?
t = Ia
m
I = m1r12 + m2r22… = mr2
Mass is distributed over entire body (from 0 to r)
Angular acceleration of every point on rigid body is equal
Moment of Inertia of a Rigid Body
What is the Moment of Inertia of a rotating rigid body?
t = Ia
I 1 = m 1r 12
t = m1r12a1 + m2r22a2…
t = (mr2)a
I = mr2
m
I2 = m2r22
Moment of Inertia of a rigid body
depends on the MASS of the object
AND the DISTRIBUTION of mass
about the AXIS of rotation
Moment of Inertia of a Rigid Body
I = mr2
r
r
F
m
F
m
Which object has a greater Moment of Inertia?
If the same force F is applied to each object as shown…
which object will have a greater angular acceleration?
Moment of Inertia of a Rigid Body
I = mr2
axis of rotation
m
axis of rotation
m
L
m
m
L
In which case is the moment of inertia of the baton greater?
If both batons were rotating with the same w, and the same
braking torque is applied to both…
Which one would come to rest sooner?
Rotational Kinetic Energy
What is the kinetic energy of a rotating object?
v
r
m
I1 = m1r12
r1
m
I 2 = m 2r 22
KE = (1/2)mv2
KEr = KEr1 + KEr2…= ∑KEr
KE = (1/2)m(rw)2
KEr = ∑(1/2)mv2
KEr =
∑(1/2)mr2w2 2 2
KEr = (1/2)(∑mr )w
KEr = (1/2)Iw2
KEr = (1/2)Iw2
Rotational Kinetic Energy
Is energy conserved as the ball rolls down the frictionless ramp?
What forms of energy does the ball have while
rolling down the ramp?
Conservation of Mechanical Energy when Wnc = 0
(KEt + KEr + PEG + PES)i = (KEt + KEr + PEG + PES)f
(1/2)mv2
mgy
(1/2)Iw2
(1/2)kx2
Rotational Kinetic Energy
Work-Energy Theorem
Wnc = DKEt + DKEr + DPEG + DPES
Conservation of Mechanical Energy when Wnc = 0
(KEt + KEr + PEG + PES)i = (KEt + KEr + PEG + PES)f
(1/2)mv2
mgy
(1/2)Iw2
(1/2)kx2
Rotational Kinetic Energy
Sphere radius = R
Is = (2/5)mR2
Cube length
= 2R
m
m
h
h
q
q
What forms of energy does each object have….
at the top of the ramp (before being released)?
halfway down the ramp?
at the bottom of the ramp?
What is the speed of each object when it reaches the bottom of
the frictionless ramp (in terms of m,g, h, R and q)?
Which object reaches the bottom first?
Angular Momentum
D(mv)
∑F = ma =
Dt
Newton’s 2nd Law
D(Iw)
∑t = Ia =
Dt
Rotational Analog to
Newton’s 2nd Law
p = mv
Dp
∑F =
Dt
Linear
Momentum
Relating F & p
L = Iw
DL
∑t =
Dt
Angular
Momentum
Relating t & L
The net TORQUE acting on an object is equal to the CHANGE in
ANGULAR MOMENTUM in a given TIME interval
Conservation of Angular Momentum
The net TORQUE acting on an object is equal to the CHANGE in
ANGULAR MOMENTUM in a given TIME interval
DL
∑t =
Dt
if ∑t = 0 then DL = 0
If NO net external TORQUE is acting on an object then
ANGULAR MOMENTUM is CONSERVED
L i = Lf
Conservation of Angular Momentum
Iiwi = Ifwf
if ∑t = 0
Angular Momentum
R
M
w1
R
w2 = ?
M
You (mass m) are standing at the center of a merry-go-round (I =
(1/2)MR2) which is rotating with angular speed w1, as you walk to
the outer edge of the merry-go-round…
What happens the angular momentum of the system?
What happens the angular speed of the merry-go-round?
What happens to the rotational kinetic energy of the system?
Practice Problem
A 10.00-kg cylindrical reel with the radius of 0.500 m and a frictionless
axle starts from rest and speeds up uniformly as a 5.00 kg bucket falls
into a well, making a light rope unwind from the reel. The bucket starts
from rest and falls for 5.00 s.
10.0 kg
0.500 m
What is the linear acceleration
of the falling bucket?
How far does it drop?
What is the angular acceleration of the reel?
5.00 kg
Use energy conservation principles to determine
the speed of the spool
after the bucket has fallen 5.00 m
Practice Problem
Two astronauts, each having a mass of 100.0-kg, are connected by a
10.0 m rope of negligible mass. They are isolated in space, moving in
circles around the point halfway between them at a speed of 5.00 m/s.
Treating the astronauts as particles…
What is the magnitude of the angular momentum and the rotational
energy of the system?
By pulling on the rope, the astronauts shorten the distance between
them to 5.00 m…
What is the new angular momentum of the system?
What are their new angular and linear speeds?
What is the new rotational energy of the system?
How much work is done by the astronauts in shortening the rope?
Questions of the Day
1) A solid sphere and a hoop of equal radius and mass are both rolled
up an incline with the same initial velocity. Which object will travel
farthest up the inclined plane?
a) the sphere
b) the hoop
c) they’ll both travel the same distance up the plane
d) it depends on the angle of the incline
2) If an acrobat rotates once each second while sailing through the air,
and then contracts to reduce her moment of inertia to 1/3 of what is
was, how many rotations per second will result?
a) once each second
b) 3 times each second
c) 1/3 times each second
d) 9 times each second