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Lecture 15
Rotational Dynamics
Reading and Review
Moment of Inertia
The moment of inertia I:
The total kinetic energy of a
rolling object is the sum of
its linear and rotational
kinetic energies:
Rolling Down
Two spheres start rolling down a ramp
from the same height at the same time.
One is made of solid gold, and the other
of solid aluminum.
Which one reaches the bottom first?
a) solid aluminum
b) solid gold
c) same
d) can’t tell without more
information
Rolling Down
Two spheres start rolling down a ramp
from the same height at the same time.
One is made of solid gold, and the other
of solid aluminum.
Which one reaches the bottom first?
a) solid aluminum
b) solid gold
c) same
d) can’t tell without more
information
initial PE: mgh
Moment of inertia depends on
mass and distance from axis
final KE:
squared. For a sphere:
I = 2/5 MR2
But you don’t need to know that!
2 cancels out!
MR
All you need to know is that it
Mass and radius don’t matter, only
depends on MR2
the distribution of mass (shape)!
Moment of Inertia
Two spheres start rolling down a ramp at
the same time. One is made of solid
aluminum, and the other is made from a
hollow shell of gold.
a) solid aluminum
b) hollow gold
c) same
Which one reaches the bottom first?
d) can’t tell without more
information
Moment of Inertia
Two spheres start rolling down a ramp at
the same time. One is made of solid
aluminum, and the other is made from a
hollow shell of gold.
a) solid aluminum
b) hollow gold
c) same
Which one reaches the bottom first?
initial PE: mgh
final KE:
Larger moment of inertia -> lower
velocity for the same energy.
d) can’t tell without more
information
A solid sphere has more of its mass
close to the center. A shell has all
of its mass at a large radius.
A shell has a larger moment of
inertia than a solid object of the
same mass, radius and shape
Power output of the Crab pulsar
•Power output of the Crab pulsar, in radio and X-rays, is about 6 x 1031 W
(which is about 150,000 times the power output of our sun). Since the
pulsar is out of nuclear fuel, where does all this energy come from ?
• The angular speed of the pulsar, and so the rotational kinetic energy,
is going down over time. This kinetic energy is converted into the
energy coming out of that star.
• calculate the rotational kinetic energy at the beginning and at the end
of a second, by taking the moment of inertia to be 1.2x1038 kg-m2 and
the initial angular speed to be 190 s-1. Δω over one second is given by
the angular acceleration.
1 2
KE i  I
2
1
1 2 1
1
2
2
KE f  I     I  I 2     I  
2
2
2
2
KE  I      I   2
Power output of the Crab pulsar
•Power output of the Crab pulsar, in radio and X-rays, is about 6 x 1031 W
(which is about 150,000 times the power output of our sun). Since the
pulsar is out of nuclear fuel, where does all this energy come from ?
• The angular speed of the pulsar, and so the rotational kinetic energy,
is going down over time. This kinetic energy is converted into the
energy coming out of that star.
• calculate the rotational kinetic energy at the beginning and at the end
of a second, by taking the moment of inertia to be 1.2x1038 kg-m2 and
the initial angular speed to be 190 s-1. Δω over one second is given by
the angular acceleration.
Torque
We know that the same force will be much more
effective at rotating an object such as a nut or a
door if our hand is not too close to the axis.
This is why we have
long-handled wrenches,
and why doorknobs are
not next to hinges.
The torque increases as the force increases,
and also as the distance increases.
Only the tangential component of force
causes a torque
A more general definition of torque:
Fsinθ
  r F
Fcosθ
Right Hand Rule
You can think of this as either:
- the projection of force on to the tangential direction
OR
- the perpendicular distance from the axis of rotation to line of the force
Torque
If the torque causes a counterclockwise angular
acceleration, it is positive; if it causes a clockwise
angular acceleration, it is negative.
Using a Wrench
You are using a wrench to
loosen a rusty nut. Which
a
b
arrangement will be the
most effective in tightening
the nut?
c
d
e) all are equally effective
Using a Wrench
You are using a wrench to
loosen a rusty nut. Which
a
b
arrangement will be the
most effective in tightening
the nut?
Because the forces are all the
same, the only difference
is the lever arm. The
arrangement with the largest
lever arm (case #2) will provide
the largest torque.
c
d
e) all are equally effective
The gardening tool shown is used to pull weeds.
If a 1.23 N-m torque is required to pull a given
weed, what force did the weed exert on the tool?
What force was used
on the tool?
Force and Angular Acceleration
Consider a mass m rotating around an axis a
distance r away.
Newton’s second law:
a=rα
Or equivalently,
Torque and Angular Acceleration
Once again, we have analogies between linear and
angular motion:
The L-shaped object shown below consists of
three masses connected by light rods. What
torque must be applied to this object to give it an
angular acceleration of 1.2 rad/s2 if it is rotated
about
(a) the x axis,
(b) the y axis
(c) the z axis (through the origin and
(a)
perpendicular to the page)
(b)
(c)
Torque
Only the tangential component of force causes a torque
Project the force onto
the tangential direction
Fsinθ
Fcosθ
  r F
  rF  rF sin
Torque and Angular Acceleration
Angular motion is analogous to linear motion
The L-shaped object shown below consists of three
masses connected by light rods. What torque must
be applied to this object to give it an angular
acceleration of 1.2 rad/s2 if it is rotated about
an axis parallel to the y axis, and through the 2.5kg
mass?
The L-shaped object shown below consists of three
masses connected by light rods. What torque must
be applied to this object to give it an angular
acceleration of 1.2 rad/s2 if it is rotated about
an axis parallel to the y axis, and through the 2.5kg
mass?
A 2.85-kg bucket is attached to a disk-shaped pulley of radius 0.121 m
and mass 0.742 kg. If the bucket is allowed to fall,
(a) what is its linear acceleration?
(b) What is the angular acceleration of the pulley?
(c) How far does the bucket drop in 1.50 s?
A 2.85-kg bucket is attached to a disk-shaped pulley of radius 0.121 m
and mass 0.742 kg. If the bucket is allowed to fall,
(a) what is its linear acceleration?
(b) What is the angular acceleration of the pulley?
(c) How far does the bucket drop in 1.50 s?
(a) Pulley spins as bucket falls
(b)
(c)
Static Equilibrium
Static equilibrium describes an object at rest –
neither rotating nor translating.
If the net torque is zero, it
doesn’t matter which axis we
consider rotation to be around;
you choose the axis of rotation
This can greatly simplify a problem
X
Center of Mass and
Gravitational Force on
an Extended Object axis of
m1
...
mj
X
xj
rotation
Fj = m j g
center of mass
m1 ... mj
xj
X
axis of
rotation
xcm
F = Mg
Balance
If an extended object is to be balanced, it must be
supported through its center of mass.
Center of Mass and Balance
This fact can be used to find the center of mass of
an object – suspend it from different axes and trace
a vertical line. The center of mass is where the lines
meet.
Balancing Rod
A 1-kg ball is hung at the end of a rod
1-m long. If the system balances at a
a) ¼ kg
b) ½ kg
point on the rod 0.25 m from the end
c) 1 kg
holding the mass, what is the mass of
d) 2 kg
the rod?
e) 4 kg
1m
1kg
Balancing Rod
A 1-kg ball is hung at the end of a rod
1-m long. If the system balances at a
a) ¼ kg
b) ½ kg
point on the rod 0.25 m from the end
c) 1 kg
holding the mass, what is the mass of
d) 2 kg
the rod?
e) 4 kg
The total torque about the pivot
must be zero !!
The CM of the
same distance
rod is at its center, 0.25 m to the
X
right of the pivot. Because this
must balance the ball, which is
the same distance to the left of
the pivot, the masses must be
the same !!
mROD = 1 kg
1 kg
CM of rod
When you arrive at Duke’s Dude Ranch, you are greeted by the large
wooden sign shown below. The left end of the sign is held in place by a bolt,
the right end is tied to a rope that makes an angle of 20.0° with the
horizontal. If the sign is uniform, 3.20 m long, and has a mass of 16.0 kg,
what is
(a) the tension in the rope, and
(b) the horizontal and vertical components of the force, exerted by the
bolt?
When you arrive at Duke’s Dude Ranch, you are greeted by the large
wooden sign shown below. The left end of the sign is held in place by a bolt,
the right end is tied to a rope that makes an angle of 20.0° with the
horizontal. If the sign is uniform, 3.20 m long, and has a mass of 16.0 kg,
what is
(a) the tension in the rope, and
(b) the horizontal and vertical components of the force exerted by the
Torque, vertical force, and horizontal force are all zero
bolt?
But I don’t know two of the forces!
I can get rid of one of them, by choosing my axis of
rotation where the force is applied.
Choose the bolt as the axis of rotation, then:
(b)
Dumbbell I
A force is applied to a dumbbell
for a certain period of time, first
as in (a) and then as in (b). In
which case does the dumbbell
acquire the greater
center-of-mass speed ?
a) case (a)
b) case (b)
c) no difference
d) it depends on the rotational
inertia of the dumbbell
Dumbbell I
A force is applied to a dumbbell
for a certain period of time, first
as in (a) and then as in (b). In
which case does the dumbbell
acquire the greater
center-of-mass speed ?
Because the same force acts for the
same time interval in both cases, the
change in momentum must be the
same, thus the CM velocity must be
the same.
a) case (a)
b) case (b)
c) no difference
d) it depends on the rotational
inertia of the dumbbell
F = ma implies Newton’s first law:
without a force, there is no acceleration
Now we have
Linear momentum was the concept that tied together Newton’s Laws,
is there something similar for rotational motion?
Angular Momentum
Consider a particle moving in a circle
of radius r,
I = mr2
L = Iω = mr2ω = rm(rω)
= rmvt = rpt
Angular Momentum
For more general motion (not necessarily circular),
The tangential component of the
momentum, times the distance
Angular Momentum
For an object of constant moment of inertia,
consider the rate of change of angular momentum
analogous to 2nd Law
for Linear Motion
Conservation of Angular Momentum
If the net external torque on a system is zero, the
angular momentum is conserved.
As the moment of inertia decreases, the
angular speed increases, so the angular
momentum does not change.
Conservation of Angular Momentum
Angular momentum is also conserved in rotational
collisions
Figure Skater
A figure skater spins with her arms
a) the same
extended. When she pulls in her arms,
b) larger because she’s rotating
she reduces her rotational inertia
faster
and spins faster so that her angular
momentum is conserved. Compared
c) smaller because her rotational
inertia is smaller
to her initial rotational kinetic energy,
her rotational kinetic energy after she
pulls in her arms must be:
Figure Skater
A figure skater spins with her arms
a) the same
extended. When she pulls in her arms,
b) larger because she’s rotating
she reduces her rotational inertia
faster
and spins faster so that her angular
momentum is conserved. Compared
c) smaller because her rotational
inertia is smaller
to her initial rotational kinetic energy,
her rotational kinetic energy after she
pulls in her arms must be:
KErot = I 2 = L2 /I (used L = I ).
Because L is conserved, smaller I
means larger KErot. The “extra”
energy comes from the work she
does on her arms.
Rotational Work
A torque acting through an angular
displacement does work, just as a force acting
through a distance does.
Consider a tangential force on
a mass in circular motion:
τ=rF
Work is force times the distance on the arc:
W=sF
s = r Δθ
W = (r Δθ) F = rF Δθ = τ Δθ
The work-energy theorem applies as usual.
Rotational Work and Power
Power is the rate at which work is done, for
rotational motion as well as for translational
motion.
Again, note the analogy to the linear form (for
constant force along motion):
The Vector Nature of Rotational Motion
The direction of the angular velocity vector is along
the axis of rotation. A right-hand rule gives the sign.
Right-hand Rule:
your fingers should
follow the velocity
vector around the
circle
The Torque Vector
Similarly, the right-hand rule gives the direction
of the torque vector, which also lies along the
assumed axis or rotation
Right-hand Rule:
your fingers should
follow the force
vector around the
circle
The linear momentum of components related to
the vector angular momentum of the system
Applied tangential force related to the torque vector
Applied torque over time related to change
in the vector angular momentum.
Cassette Player
When a tape is played on a cassette
deck, there is a tension in the tape
that applies a torque to the supply
reel. Assuming the tension remains
constant during playback, how does
this applied torque vary as the supply
reel becomes empty?
a) torque increases
b) torque decreases
c) torque remains constant
Cassette Player
When a tape is played on a cassette
deck, there is a tension in the tape
that applies a torque to the supply
reel. Assuming the tension remains
constant during playback, how does
this applied torque vary as the supply
reel becomes empty?
As the supply reel empties, the lever
arm decreases because the radius of
the reel (with tape on it) is
decreasing. Thus, as the playback
continues, the applied torque
diminishes.
a) torque increases
b) torque decreases
c) torque remains constant