Download Lecture 19

Physics I
95.141
LECTURE 19
11/17/10
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
Exam Prep Problem
• A ball (ball A) of mass m=2kg, traveling at a velocity vA=4m/s.
collides with two balls at rest. Ball B (mB=4kg) leaves the collision at
an angle of +45 degrees, and ball C (mC=2kg) leaves the collision at
an angle of -45 degrees. Ball A is at rest after the collision.
– A) (5pts) Write down the conservation of momentum expressions for
this collision
– B) (10pts) Solve your system of equations to determine the velocities of
balls B and C after the collision
– C) (10pts) Is this an elastic or inelastic collision? Explain why. If it is
inelastic, how much thermal energy is generated in the collision?
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
Exam Prep Problem
• mA=mC=2, vA=4m/s vB=vC=0, mB=4
• θ’B=+45°, θ’C=-45°, v’A=0m/s
– A) (5pts) Write down the conservation of momentum
expressions for this collision
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
Exam Prep Problem
• mA=mC=2, vA=4m/s vB=vC=0, mB=4
• θ’B=+45°, θ’C=-45°, v’A=4m/s
•
B) (10pts) Solve your system of equations to determine the
velocities of balls B and C after the collision
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
Exam Prep Problem
• mA=mC=2, vA=4m/s vB=vC=0, mB=4
• θ’B=+45°, θ’C=-45°, v’A=4m/s
•
C) (10pts) Is this an elastic or inelastic collision? Explain
why. If it is inelastic, how much thermal energy is generated
in the collision?
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
Outline
•
•
•
Torque
Rotational Inertia
Moments of Inertia
•
What do we know?
– Units
– Kinematic equations
– Freely falling objects
– Vectors
– Kinematics + Vectors = Vector
Kinematics
– Relative motion
– Projectile motion
– Uniform circular motion
– Newton’s Laws
– Force of Gravity/Normal Force
– Free Body Diagrams
– Problem solving
– Uniform Circular Motion
– Newton’s Law of Universal Gravitation
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
Weightlessness
Kepler’s Laws
Work by Constant Force
Scalar Product of Vectors
Work done by varying Force
Work-Energy Theorem
Conservative, non-conservative Forces
Potential Energy
Mechanical Energy
Conservation of Energy
Dissipative Forces
Gravitational Potential Revisited
Power
Momentum and Force
Conservation of Momentum
Collisions
Impulse
Conservation of Momentum and Energy
Elastic and Inelastic Collisions2D, 3D
Collisions
Center of Mass and translational motion
Angular quantities
Vector nature of angular quantities
Constant angular acceleration
Review of Lecture 18
• Discussed angular quantities we use to describe
rotational motion
– Angular Displacement:
– Angular Velocity:
– Angular Acceleration:
 [rad ]
 [ rad s ]
 [rad s ]
2
• Vector Nature of Angular Velocity
• Rotation with constant angular acceleration, and
parallels with constant linear acceleration problems.
1 2
   o   o t  t
2
    2    o 
2
2
o
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
1 2
x  xo  vo t  at
2
v 2  vo2  2a ( x  xo )
Example
• A top is brought up to speed with α=7rad/s2 in
1.5s. After that it slows down slowly with α=0.1rad/s2 until it stops spinning.
– A) What is the fastest angular velocity of the top?
– B) How long does it take the top to stop spinning once
it reaches its top angular velocity?
– C) How many rotations does the top make in this time?
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
Example
• A top is brought up to speed with α=7rad/s2 in
1.5s. After that it slows down slowly with α=0.1rad/s2 until it stops spinning.
– C) How many rotations does the top make in this time?
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
More Angular Quantities
• We have discussed the angular equivalents for
position, velocity, and acceleration (Chapter 2).
• But is this where the similarities end?
• After we discussed linear motion, we discussed
the Forces that cause this motion.
• Is there an equivalent to Force for rotational
motion?
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
Torque
• Clearly it takes a Force to make something start
rotating, but is the magnitude/direction of the
Force the only thing that matters?
F
F
• Think about opening a door. Where is it easier
to push the door open, near the hinges, or near
the handle?
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
Torque
• The effect of a Force on the rotational object
depends on the perpendicular distance from the
axis of rotation that the Force is applied.
• This distance is known as the lever arm or
moment arm.
F
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
Torque
• Angular acceleration is proportional to the
product of the Force and the lever arm (torque).
 
• But its not just the total Force, of course, life’s
not that easy….
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
Torque
• The Torque can be defined as the product of the
lever arm and the component of the Force
perpendicular to the lever arm.
  RF

F
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
Example
• Two forces are applied to compound wheel as
shown below. What is the net Torque on the
object? (RA=0.3m, RB=0.5m)
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
Rotational Dynamics
• We know that for linear acceleration, Newton’s
2nd Law tells us that the linear acceleration is
proportional to Force.

 
F a

F  ma
• For rotational motion, we know the kinematic
equations are similar to those for linear motion.
So angular acceleration is proportional to the
rotational equivalent of Force (Torque). What is
the rotational equivalent of mass/inertia.


 
95.141, F2010, Lecture 19
Department of Physics and Applied Physics


  ?
Rotational Dynamics
• If we have a point mass a distance R from an axis of
rotation, and we apply a Force perpendicular to R, what
is acceleration?


F  ma
a  R
  RF
• The quantity mR2 is known
as the rotational inertia of
the particle: moment of
inertia.
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
Calculating Moments of Inertia
• Say we have two masses connected to a
massless rod.
4m
1m
3kg
A
B 2kg
5m
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
Rotational Dynamics
• What if we have, instead of a point source, a
solid object?
• We could divide the solid object into a large
number of smaller masses dm, and calculate the
rotational inertia of each of these….
2



m
R
  i i 
2
2
2
2
m
R

m
R

m
R

m
R
 i i 1 1 2 2 3 3 ...  I
  I
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
 F  Ma
I for a solid object
• Thin hoop
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
I for a solid object
• Circular Plate
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
I for a solid object
• Rod
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
Common Moments of Inertia
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
Example Problem
Mp=10kg
R
• A mass of 10kg is attached to a cylindrical
pulley of radius R1 and mass mp=10kg and
released from rest. What is the acceleration of
the mass?
Mb=10kg
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
Example Problem
• What is the angular acceleration of the rod shown below,
if it is released from rest, at the moment it is released?
What is the linear acceleration of the tip?
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
WIPEOUT
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
Parallel Axis Theorem
• What is the CM for the system we looked at earlier?
3kg
A
B 2kg
5m
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
Parallel Axis Theorem
CM
3kg
2kg
h
5m
• Says that, for rotation about an axis h from the CM
I  I CM  Mh 2
• What is ICM? I
 35kgm 2
I
 50kgm 2
CM 1
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
CM  2
Parallel Axis Theorem Example
• What is the moment of inertia for a rod
– Rotating about its center of mass?
– Rotating about its end?
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
Rotational Kinetic Energy
• We now know the rotational equivalent of mass
is the moment of inertia I.
• If I told you there was such a thing as rotational
kinetic energy, you could probably make a good
guess as to what form it would take…
1
KEtrans  mv 2
2
KErot
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
1 2
 I
2
Rotational Kinetic Energy
• Prove it!
1
KEtrans   mv 2
2
95.141, F2010, Lecture 19
Department of Physics and Applied Physics
v  R