Transcript Department of Physics and Applied Physics 95.141, S2010, Lecture 23

Physics I
95.141
LECTURE 23
5/10/10
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
Exam Prep Question
•
A mass of 1kg is attached to a vertical
spring. The spring deflects 2cm.
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a) (10 pts) What is the spring constant k
of the spring?
b) (10 pts) A 50g bullet is shot at 100m/s
from below into the mass, and ends
embedded in the mass. What is the
velocity of the mass/bullet after the
collision?
c) (5pts) What is the new equilibrium
position of the spring/mass system after
the collision?
d) (5pts) What is the total energy of the
spring/mass system immediately after the
collision? (remember, the system has a
new mass now, so it will have a new
equilibrium position)
e) (5pts) What is the amplitude of
oscillation of the spring mass system after
the collision?
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
k
m=1kg
v=500m/s
m=50g
Exam Prep Question
•
A mass of 1kg is attached to a vertical
spring. The spring deflects 2cm.
•
a) (10 pts) What is the spring constant k
of the spring?
k
m=1kg
v=500m/s
m=50g
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
Exam Prep Question
•
A mass of 1kg is attached to a vertical
spring. The spring deflects 2cm.
•
b) (10 pts) A 50g bullet is shot at 100m/s
from below into the mass, and ends
embedded in the mass. What is the
velocity of the mass/bullet after the
collision?
k
m=1kg
v=500m/s
m=50g
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
Exam Prep Question
•
A mass of 1kg is attached to a vertical
spring. The spring deflects 2cm.
•
c) (5pts) What is the new equilibrium
position of the spring/mass system after
the collision?
k
m=1kg
v=500m/s
m=50g
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
Exam Prep Question
•
A mass of 1kg is attached to a vertical
spring. The spring deflects 2cm.
•
d) (5pts) What is the total energy of the
spring/mass system immediately after the
collision? (remember, the system has a
new mass now, so it will have a new
equilibrium position)
k
m=1kg
v=500m/s
m=50g
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
Exam Prep Question
•
A mass of 1kg is attached to a vertical
spring. The spring deflects 2cm.
•
e) (5pts) What is the amplitude of
oscillation of the spring mass system after
the collision?
k
m=1kg
v=500m/s
m=50g
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
Administrative Notes
• Physics I Final:
– SATURDAY 5/15/10
– Olney 150 (HERE)
– 3:00 P.M.
• 8 total problems, 1 multiple choice
• Extra Time: Starts at 12:00 pm
– Meet at my office
• Review Session Thursday (5/13), 6:30 pm,
OH218.
• 20 problems posted on-line. 5 will be on the
Final.
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
Outline
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Pendulums
Damped and Forced Harmonic Motion
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
– Weightlessness
– Kepler’s Laws
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
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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
Torque
Rotational Inertia
Moments of Inertia
Angular Momentum
Vector Cross Products
Conservation of Angular Momentum
Oscillations
Simple Harmonic Motion
Review of Lecture 22
• Discussed, qualitatively, oscillatory motion of spring
mass system: shifting of energy between elastic potential
energy (spring) and kinetic energy (mass)
• Quantitative description of motion of an object with
constant restoring force
2
d x (t )
 kx (t )  m
dt 2
• Developed description of motion of spring mass from the
differential equation x(t )  A cos(t   ) ,   k m
• Used this to determine velocity and acceleration
v (t )   A sin( t   )
functions
2
a
(
t
)


A

cos(t   )
• Energy of a SHO
1
1
1 2 1
2
2
2
E

k
(
x
(
t
))

m
(
v
(
t
))

kA

mv
total
max
95.141, S2010, Lecture
23
2
2
2
2
Department of Physics and Applied Physics
The pendulum
• A simple pendulum consists of a mass
(M) attached to a massless string of
length L.
• We know the motion of the mass, if
dropped from some height, resembles
simple harmonic motion: oscillates
back and forth.
• Is this really SHO? Definition of SHO
is motion resulting from a restoring
force proportional to displacement.
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
Simple Pendulum
L
• We can describe displacement
as:
• The restoring Force comes from
gravity, need to find component of
force of gravity along x
• Need to make an approximation
here for small θ…
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
θ
Δx
Simple Pendulum
• Now we have an expression for
the restoring force
F  mg sin   mg 
L
θ
x  L
mg
x
L
• From this, we can determine the
effective “spring” constant k
F 
• And we can determine the natural
frequency of the pendulum
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
Δx
Simple Pendulum
• If we know

g
L
L
θ
• We can determine period T
• And we can the equation of
motion for displacement in x
• …or θ
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
Δx
Damped Harmonic Motion
• If I let the pendulum swing, would it keep
returning to the same original displacement?
• In the real world there are other forces, in
addition to the restoring force which act on the
pendulum (or any oscillator).
• The harmonic motion for these real-world
oscillators is no longer simple.
• Damped Harmonic Motion
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
Damped Harmonic Motion
• Suppose there is a damping force acting on the
oscillator which depends on velocity
– This is a Force which acts against the oscillator,
opposite the direction of motion.
Fdamping
dx
 bv  b
dt
• The force equation now looks like:
ma  kx  bv
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
Damped Harmonic Motion
• The solution to this differential equation is
trickier, but let’s try the following solution:
x (t )  Ae t cos  t
k
b2
 

m 4m 2
b

2m
• Natural frequency decreases
• Amplitude of oscillations decreases
exponentially.
95.141, S2010, Lecture 23
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Simple Harmonic Oscillation
2
k
o 
m
k  400 N m
m  2kg
1
x(t)
x ( t )  A cos o t
0
-1
-2
0
1
2
3
4
Time (s)
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
5
6
Damped Harmonic Oscillation
2
k
b2
 

m 4m 2
k  400 N m
m  2kg
b  2 Ns m
1
x(t)
x (t )  A cos  t
0
-1
-2
0
1
2
3
4
Time (s)
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
5
6
Damped Harmonic Oscillation
x ( t )  Ae t
k
b2
 

m 4m 2
k  400 N m
m  2kg
b  2 Ns m
b

2m
2
1
x(t)
x ( t )  A cos  t
0
-1
-2
0
1
2
3
4
Time (s)
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
5
6
Damped Harmonic Oscillation
k
b2
 

m 4m 2
k  400 N m
m  2kg
b  2 Ns m
b

2m
2
1
x(t)
x ( t )  Ae t cos  t
0
-1
-2
0
1
2
3
4
Time (s)
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
5
6
Forced Harmonic Motion
• In addition to damping, one can apply a force to
an oscillator. If that external force is sinusoidal,
the Force equation looks like:
ma  Fo cos t  bv  kx
o 
d2x
dx
m 2  b  kx  Fo cos t
dt
dt
k
m
• The solution to this differential equation is:
x  Ao sin( t  o ) Ao 
Fo
m ( 2  o2 ) 2  b 
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
2
 2  o2
o  tan
(b m)
1
2
m2
Forced Harmonic Motion
Ao 
m ( 2  o2 ) 2  b 
Fo  2 N
b  2 Ns m
0.30
0.25
Amplitude (Ao)
Fo
0.20
0.15
0.10
0.05
0.00
0
5
10
 (rad/s)
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
15
20
2
2
m2
Forced Harmonic Motion
Ao 
m ( 2  o2 ) 2  b 
Fo  2 N
b  1 Ns m
0.30
0.25
Amplitude (Ao)
Fo
0.20
0.15
0.10
0.05
0.00
0
5
10
 (rad/s)
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
15
20
2
2
m2
Forced Harmonic Motion
Ao 
m ( 2  o2 ) 2  b 
Fo  2 N
b  0.5 Ns m
0.30
0.25
Amplitude (Ao)
Fo
0.20
0.15
0.10
0.05
0.00
0
5
10
 (rad/s)
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
15
20
2
2
m2
In the real world?
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
Waves (Chapter 15)
• A wave is a displacement that travels (almost
always through a medium) with a velocity and
carries energy.
– It is the displacement that travels, not the medium!!
– The wave travels over large distances, the displacement
is small compared to these distances.
– All forms of waves transport energy
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
Waves (Water Waves)
• Example which most frequently comes to mind
are waves on the ocean.
– With an ocean wave, it is not the water that is
travelling with the lateral velocity.
– Water is displaced up and down
– This displacement is what moves!
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
Waves (Earthquakes)
• Earthquakes are waves where the displacement
is of the surface of the Earth.
– Again, the Earth’s surface is not travelling with any
lateral velocity. It is the displacement which travels.
– The surface of the Earth moves up and down.
– Obviously a lot of Energy is transported!
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
Waves (Sound Waves)
• Sound is also a form of wave.
– The displacement for a sound wave is not an “up and
down” displacement. It’s a compression.
– The air is compressed, and it is the compression
which travels through air.
– Sound is not pockets of compressed air travelling, but
the compression of successive portions of air.
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
Waves (Light)
• Light is also a type of wave
– The displacement of a light wave is a change in the
Electric Field.
– This propagates through space with the speed of light
c  310
8m
s
– Light can carry energy:
• Solar power
• Radiative heating
• Lasers
– Green lasers can be especially damaging to the eyes, since our
eyes are most sensitive to green light.
95.141, S2010, Lecture 23
Department of Physics and Applied Physics
Characteristics of Waves
• A continuous or periodic wave has a source which is
continuous and oscillating
– Think of a hand oscillating a piece of rope up and down
– Or a speaker playing a note
• This vibration is the source of the wave, and it is the
vibration that propagates.
• If we freeze that wave in time (take a picture)
x
95.141, S2010, Lecture 23
Department of Physics and Applied Physics