Transcript Lecture14-10

Lecture 14
Rotational Kinematics
Reading and Review
Rotational Motion
Arc length s, measured in radians:
Analogies between linear and
rotational kinematics:
s = rθ
Linear and Angular Velocity
Bonnie and Klyde
Bonnie sits on the outer rim of a
merry-go-round, and Klyde sits
midway between the center and the
rim. The merry-go-round makes one
revolution every 2 seconds. Klyde and
Bonnie have about the same mass.
Who has the larger kinetic energy?
a) Klyde
b) Bonnie
c) both the same
d) the net kinetic energy for
both of them is zero since
there is no average
motion of center of mass

Klyde
Bonnie
Bonnie and Klyde
Bonnie sits on the outer rim of a
merry-go-round, and Klyde sits
midway between the center and the
rim. The merry-go-round makes one
revolution every 2 seconds. Klyde and
Bonnie have about the same mass.
Who has the larger kinetic energy?
a) Klyde
b) Bonnie
c) both the same
d) the net kinetic energy for
both of them is zero since
there is no average
motion of center of mass
Their linear speeds v will be

different because v = r  and
Bonnie is located farther out
(larger radius r) than Klyde.
Therefore KEBonnie > KEKlyde
Klyde
Bonnie
Rolling Motion
We may also consider rolling motion to be a
combination of pure rotational and pure
translational motion:
+
=
Rolling Motion
If a round object rolls without slipping, there is a
fixed relationship between the translational and
rotational speeds:
Truck Speedometer
Suppose that the speedometer of a
truck is set to read the linear speed
a)
of the truck but uses a device that
actually measures the angular
speed of the tires. If larger
b)
diameter tires are mounted on the
truck instead, how will that affect
the speedometer reading as
c)
compared to the true linear speed
of the truck?
speedometer reads a higher
speed than the true linear speed
speedometer reads a lower speed
than the true linear speed
speedometer still reads the true
linear speed
Truck Speedometer
Suppose that the speedometer of a
truck is set to read the linear speed
a)
of the truck but uses a device that
actually measures the angular
speed of the tires. If larger
b)
diameter tires are mounted on the
truck instead, how will that affect
the speedometer reading as
c)
compared to the true linear speed
of the truck?
speedometer reads a higher
speed than the true linear speed
speedometer reads a lower speed
than the true linear speed
speedometer still reads the true
linear speed
The linear speed is v =  R. So when the speedometer measures
the same angular speed  as before, the linear speed v is actually
higher, because the tire radius is larger than before.
Jeff of the Jungle swings on a vine that is 7.20 m long. At the
bottom of the swing, just before hitting the tree, Jeff’s linear
speed is 8.50 m/s.
(a) Find Jeff’s angular speed at this time.
(b) What centripetal acceleration does Jeff experience at the
bottom of his swing?
(c) What exerts the force that is responsible for Jeff’s
centripetal acceleration?
Jeff of the Jungle swings on a vine that is 7.20 m long. At the
bottom of the swing, just before hitting the tree, Jeff’s linear
speed is 8.50 m/s.
(a) Find Jeff’s angular speed at this time.
(b) What centripetal acceleration does Jeff experience at the
bottom of his swing?
(c) What exerts the force that is responsible for Jeff’s
centripetal acceleration?
a)
b)
c)
This is the force that is responsible for
keeping Jeff in circular motion: the vine.
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:
I   MR
2
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.
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 decreasing 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 decreasing 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 decreasing 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 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.
Rotational Dynamics
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)
Provided in lecture
notes on: 10/19
PHYS2010 midterm 2, Fall 2008
4) Two uniform solid spheres have the same mass, but one has twice the radius of the
other. The ratio of the larger sphere's moment of inertia to that of the smaller sphere is
A) 8/5
B) 4
C) 2
D) 1/2
E) 4/5
suggested time: 1-2 minutes
Please do not ask questions about this problem at
discussion sessions before 10/21
Provided in lecture
notes on: 10/19
PHYS2010 midterm 2, Fall 2008
7) A solid disk is released from rest and rolls without slipping down an inclined plane
that makes an angle of 25.0o with the horizontal. What is the speed of the disk after it
has rolled 3.00 m, measured along the plane? (Moment of inertia of a solid disk of
mass M and radius R is 1/2(MR2)).
A) 5.71 m/s
B) 2.04 m/s
C) 3.53 m/s
D) 4.07 m/s
E) 6.29 m/s
suggested time: 4-5 minutes
Please do not ask questions about this problem at
discussion sessions before 10/21