Transcript Lecture-14-10

Lecture 14
Rotational Kinematics
Office Hours:
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Rolling Motion
If a round object rolls without slipping, there is a
fixed relationship between the translational and
rotational speeds:
Rolling Motion
We may also consider rolling motion to be a
combination of pure rotational and pure
translational motion:
+
=
Rolling Wheel
A wheel rolls without slipping.
Which vector best represents
the velocity of point A?
a)
b)
c)
d)
e) the velocity at
point A is zero
A
Rolling Wheel
A wheel rolls without slipping.
Which vector best represents
the velocity of point A?
b)
c)
d)
e) the velocity at
point A is zero
A
+
a)
v
vtrans
vrot
Rotational Kinetic Energy
For this mass,
Rotational Kinetic Energy
For these two masses,
Ktotal = K1 + K2 = mr2ω2
Moment of Inertia
We can also write the kinetic energy as
Where I, the moment of inertia, is given by
What is the moment of inertia for two equal
masses on the ends of a (massless) rod,
spinning about the center of the rod?
Moment of Inertia
We can also write the kinetic energy as
Where I, the moment of inertia, is given by
What is the moment of inertia for a uniform ring of mass M and
radius R, rolling around the center of the ring?
Moment of Inertia for various shapes
Moments of inertia of various objects can be
calculated:
This is a concept
and will not be on
the formula sheet!
and so one can calculate the kinetic
energy for rotational motion:
A block of mass 1.5 kg is attached to a string that is
wrapped around the circumference of a wheel of
radius 30 cm and mass 5.0 kg, with uniform mass
density. Initially the mass and wheel are at rest, but
then the mass is allowed to fall.
What is the velocity of the
mass after it falls 1 meter?
A block of mass 1.5 kg is attached to a string that is
wrapped around the circumference of a wheel of
radius 30 cm and mass 5.0 kg, with uniform mass
density. Initially the mass and wheel are at rest, but
then the mass is allowed to fall.
What is the velocity of the
mass after it falls 1 meter?
Energy of a rolling object
The total kinetic energy of a rolling object is the
sum of its linear and rotational kinetic energies:
Since velocity and angular velocity are related for rolling
objects, the kinetic energy of a rolling object is a multiple
of the kinetic energy of translation.
Conservation of Energy
An object rolls down a ramp - what
is its translation and rotational
kinetic energy at the bottom?
Conservation of Energy
An object rolls down a ramp - what
is its translation and rotational
kinetic energy at the bottom?
From conservation of energy:
Note: no dependence on mass,
only on distribution of mass
Velocity at any height:
Conservation of Energy
If these two objects, of the same radius,
are released simultaneously, which will
reach the bottom first?
The disk will reach the bottom first – it
has a smaller moment of inertia. More
of its gravitational potential energy
becomes translational kinetic energy,
and less rotational.
Conservation of Energy
If these two objects, of the same radius,
are released simultaneously, which will
reach the bottom first?
What if they have different radii?
mR2 cancels, leaving
only the geometric factor
(in this case, 1/2 vs. 1)
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 shell has a larger moment of
inertia than a solid object of the
same mass, radius and shape
A solid sphere has more of its mass
close to the center. A shell has all
of its mass at a large radius. Total
mass and radius cancel in
expression for fraction of K tied up
in angular (rolling) motion.
ConcepTest
A ball is released from rest on a no-slip
surface, as shown. After reaching its
lowest point, the ball begins to rise again,
this time on a frictionless surface as
shown in the figure. When the ball
reaches its maximum height on the
frictionless surface, it is:
A) at a greater height as when it was released.
B) at a lesser height as when it was released.
• at the same height as when it was released.
• impossible to tell without knowing the mass of the ball.
• impossible to tell without knowing the radius of the ball.
ConcepTest
A ball is released from rest on a no-slip
surface, as shown. After reaching its
lowest point, the ball begins to rise again,
this time on a frictionless surface as
shown in the figure. When the ball
reaches its maximum height on the
frictionless surface, it is:
A) at a greater height as when it was released.
B) at a lesser height as when it was released.
• at the same height as when it was released.
• impossible to tell without knowing the mass of the ball.
• impossible to tell without knowing the radius of the ball.
Q: What if both sides of the half-pipe were no-slip?
The pulsar in the Crab nebula was created by a supernova explosion that
was observed on Earth in a.d. 1054. Its current period of rotation (33.0
ms) is observed to be increasing by 1.26 x 10-5 s per year.
•What is the angular acceleration of the pulsar in rad/s2 ?
•Assuming the angular acceleration of the pulsar to be constant, how
many years will it take for the pulsar to slow to a stop?
•Under the same assumption, what was the period of the pulsar when it
was created?
The pulsar in the Crab nebula was created by a supernova explosion that
was observed on Earth in a.d. 1054. Its current period of rotation (33.0
ms) is observed to be increasing by 1.26 x 10-5 s per year.
•What is the angular acceleration of the pulsar in rad/s2 ?
•Assuming the angular acceleration of the pulsar to be constant, how
many years will it take for the pulsar to slow to a stop?
•Under the same assumption, what was the period of the pulsar when it
was created?
The pulsar in the Crab nebula was created by a supernova explosion that
was observed on Earth in a.d. 1054. Its current period of rotation (33.0
ms) is observed to be increasing by 1.26 x 10-5 s per year.
•What is the angular acceleration of the pulsar in rad/s2 ?
•Assuming the angular acceleration of the pulsar to be constant, how
many years will it take for the pulsar to slow to a stop?
•Under the same assumption, what was the period of the pulsar when it
was created?
(a)
The pulsar in the Crab nebula was created by a supernova explosion that
was observed on Earth in a.d. 1054. Its current period of rotation (33.0
ms) is observed to be increasing by 1.26 x 10-5 s per year.
•What is the angular acceleration of the pulsar in rad/s2 ?
•Assuming the angular acceleration of the pulsar to be constant, how
many years will it take for the pulsar to slow to a stop?
•Under the same assumption, what was the period of the pulsar when it
was created?
(b)
(c)
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 change in rotational kinetic energy from the beginning to
the end of a second, by taking the moment of inertia to be 1.2x1038 kgm2 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 change in rotational kinetic energy from the beginning to
the end of a second, by taking the moment of inertia to be 1.2x1038 kgm2 and the initial angular speed to be 190 s-1. Δω over one second is
given by the angular acceleration.
Lecture 14
Rotational Dynamics
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:
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
Fcosθ
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
a
b
tighten a rusty nut. Which
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
a
b
tighten a rusty nut. Which
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 b) 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)
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?
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
Because
the same
force acts for the
dumbbell
acquire
same time interval in both cases, the
the greater
change in momentum must be the
same,
thus the CM velocity must be
center-of-mass
the same.
speed ?
a) case (a)
b) case (b)
c) no difference
d) it depends on the rotational
inertia of the dumbbell