Physics C Rotation Name:______________________

Physics C

Rotation

Name:_________________
_____

11/14/2013

1

Bertrand

Introducing the Linear/Angular Analog

Ta
ble
:

You should record analogous concepts
for linear and rotational motion in this table.


Displacement (Linear, Angular, Bridge)


Record in Analogs Table


Speed and Velocity (Linear, Angular, Bridge)


Record in Ana
log Table



Which calculus relationship

do you

use to

get
velocity from position?



Which calculus relationship
do you
use to
get
angular velocity from angle?



How do you determine the direction of angular
velocity?



Acceleration

(Linear, Angular, Bridge
)


Record in Analogs Table


Which calculus relationship do you use to get
acceleration from velocity or position?



Which calculus relationship do you use to get
angular acceleration from angular velocity or
angle?



How do you determine the direction of a
ngular
velocity?




Problem:

Assume the particle is speeding up.



What is the direction of the
instantaneous velocity, v?



What is the direction of the angular
velocity,

?



What is the direction of the tangential
acceleration, a
T
?



What is the direction of the angular
acceleration

?



What is the direction of the centripetal
acceleration, a
c
?



What is the direction of the overall
acceleration, a, of the particle?

First Kinem
atic Equation


Record in Analogs Table


Second Kinematic Equation


Record in Analogs Table


Third Kinematic Equation


Record in Analogs Table


P
roblem
:
The Beatle’s White Album is spinning
at 33 1/3 rpm when the power is turned off. If it
takes 1/2 minute
for the album’s rotation to stop,
what is the angular acceleration of the
phonograph album?












P
roblem
:

The angular velocity of a flywheel is
described by the equation


= (8.00
–

2.00 t 2).
Determine the angular displacement when the
flywheel reverses its direction.






Inertia
(Linear, Angular, Bridge)


Record in Analogs Table


Problem
:

A 2.0
-
kg mass and a 3.0
-
kg mass are
mounted on opposite ends a 2.0
-
m long rod of
n
egligible mass. What is the rotational inertia
about the center of the rod and about each mass,
assuming the axes of rotation are perpendicular
to the rod?

Physics C


Rotation


11/14/2013

2

Bertrand

Kinetic Energy

(Linear, Rotational, Bridge)


Record in Analogs Table


P
roblem
:

A 3.0 m long lightwe
ight rod has a 1.0
kg mass attached to one end, and a 1.5 kg mass
attached to the other. If the rod is spinning at 20
rpm about its midpoint around an axis that is
perpendicular to the rod, what is the resulting
rotational kinetic energy? Ignore the mass o
f the
rod.











Rotational Inertia Calculations


Formula for system of particles:





Formula for solid object:




Parallel axis theorem:





P
roblem
:

A solid ball of mass 300 grams and
diameter 80 cm is thrown at 28 m/s. As it travels
through

the air, it spins with an angular speed of
110 rad/second. What is its


translational kinetic energy?







rotational kinetic energy?







total kinetic energy?






Problem
:
Derive the rotational inertia of a long
thin rod of length L and mass M
about a point
1/3 from one end


using integration of I =


r2 dm















using the parallel axis theorem and the
rotational inertia of a rod about the center.








Problem
:

Derive the rotational inertia of a ring
of mass M and radius R abou
t the center using
the formula I =


r2 dm.
















Equilibrium

definition(s)

Physics C


Rotation


11/14/2013

3

Bertrand

Conditions for equilibrium:







Force (Linear, Rotational, Bridge)


Record in Analogs Table


Calculating Torque

using Vector Math





Practice Problem
What must F be to achieve equilibrium? Assume
there is no friction on the pulley axle.
10 kg
F
3 cm
2 cm
2 kg







Newton’s 2
nd

Law


Record in Analogs Table


Problem
:

A 1.0
-
kg wheel of 25
-
cm radius is at
rest on a fixed axis. A force of 0.45 N is applied
tangent to the rim of the wheel for 5 seconds.


After this time, what is the angular velocity of
the wheel?







Through what angle does the wheel rotate during
this 5 second period?

P
roblem
:

Derive an expression for the
acceleration of a flat disk of mass M and radius
R that rolls without slipping down a ramp of
angle

.













Practice problem
Calculate initial angular acceleration of rod
of mass M and length L.
Calculate initial acceleration of end of rod.






Sample problem
Calculate acceleration.
Assume pulley has mass M, radius R, and
is a uniform disk.
m
2
m
1

Physics C


Rotation


11/14/2013

4

Bertrand

Practice Problem
What is the acceleration of this system, and the magnitude of te
nsions T
1
and T
2
?
Assume the surface is frictionless, and pulley has the rotationa
l inertia of a
uniform disk.
m
1
= 2.0 kg
m
2
= 1.5 kg
m
pulley
= 0.45 kg
r
pulley
= 0.25 m
30
o
T
1
T
2








Work
(Linear, rotation, bridge)


Record in Analogs Table


How does work affect energy?





Power

(Linear, rotation, bridge)


Record i
n Analogs Table


Problem
:
A rotating flywheel provides power to
a machine. The flywheel is originally rotating at
of 2,500 rpm. The flywheel is a solid cylinder of
mass 1,250 kg and diameter of 0.75 m. If the
machine requires an average power of 12 kW,
for

how long can the flywheel provide power?


Problem
:

A uniform rod of mass M and length
L rotates around a pin through one end. It is
released from rest at the horizontal position.
What is the angular speed when it reaches the
lowest point?






What is t
he linear speed of the lowest point of
the rod at this position?






Problem
:
Text problem # 44.






Rolling without slipping

What bridge relationship applies with rolling
without slipping?



What full conservation of energy equation would
we use for an

object that rolls down an incline?





Problem
:

A solid sphere of mass M and radius
R rolls from rest down a ramp of length L and
angle

. Use Conservation of Energy to find the
linear acceleration and the speed at the bottom of
the ramp.

Physics C


Rotation


11/14/2013

5

Bertrand


Problem
:

A solid sphere of mass M and radius
R rolls from rest down a ramp of length L and
angle

. Use Rotational Dynamics to find the
linear accelera
tion and the speed at the bottom of
the ramp.















Problem
:

A hollow sphere (mass 1.5 kg, radius
2.0 m) rolls without slipping down a ramp
angled at 33
o
. After it suffers a drop of 1.0 meter
in vertical height


what is its translational speed?








what is its angular speed?







Momentum
(Linear, Angular, Bridge)


Record in Analogs Table


Problem
:
Determine the angular momentum for
the revolution of

the earth about the sun.








the moon about the earth.


Practice Problem
Determine the angular momentum for the 2 kg particle shown
a)
about the origin.
b)
about x = 2.0.
y (m)
x (m)
5.0
v = 3.0 m/s
-
5.0
5.0



Problem
:
Set up the calculation of the angular
momentum for the rotation of the earth on its
axis.














The Law of Conservation of
Momentum






The Law of Conservation of
Angular Momentum






Problem
:

A figure skater is spinning at angular
ve
to his body and reduces his rotational inertia to ½
its original value. What happens to his angular
velocity?


Physics C


Rotation


11/14/2013

6

Bertrand

Problem
:

A planet of mass m revolves around a
star of mass M in a highly elliptical orbit. At
poin
t A, the planet is 3 times farther away from
the star than it is at point B. How does the speed
v of the planet at point A compare to the speed at
point B?















Problem
:
A 50.0 kg child runs toward a 150
-
kg
merry
-
go
-
round of radius 1.5 m, and jump
s
aboard such that the child’s velocity prior to
landing is 3.0 m/s directed tangent to the
circumference of the merry
-
go
-
round. What will
be the angular velocity of the merry
-
go
-
round if
the child lands right on its edge?














Momentum and Torque:


Record in Analogs Table


Problem
:

A pilot is flying a propeller plane and
the propeller appears to be rotating clockwise as
the pilot looks at it. The pilot makes a left turn.
Does the plane “nose up” or “nose down” as the
plane turns left?

Physics C


Rotation


11/14/2013

7

Bertrand