Post on 05-Jan-2016
Physics 1B03summer - Lecture 7
Test 2
Tuesday June 2 at 9:30 am
CNH-104
Energy, Work, Power, Simple Harmonic Motion
Class will start at 11am
Physics 1B03summer - Lecture 7
Energy of a SHO
Physics 1B03summer - Lecture 7
Energy of a SHO
Recall that for a spring:
ETot = K + U = 1/2mv2 + 1/2kx2
And we know that: x=A cos(ωt+φ)
v=-A ω sin(ωt+φ)
Physics 1B03summer - Lecture 7
Energy in SHM
MLook again at the block & spring
energy) mechanical (totalconstant a
)(cos)(sin
)(cos)(sin
!
)(cos
)(sin
221
22221
222221
22212
21
222212
21
kA
ttkA
tktmAUK
k
tkAkxU
tAmmvK
We could also write E = K+U = ½ m(vmax )2
Hence: ETot = ½kA2
Physics 1B03summer - Lecture 7
Suppose you double the amplitude of the motion, whathappens to the total energy?
a) Doublesb) 4 x Largerc) Doesn’t change
Quiz
Physics 1B03summer - Lecture 7
a) Doublesb) 4 x Largerc) Doesn’t change
Quiz
Suppose you double the amplitude of the motion, whathappens to the maximum speed ?
Physics 1B03summer - Lecture 7
Suppose you double the mass and amplitude of the object, what happens to the maximum acceleration ?
a) Doublesb) 4 x Largerc) Doesn’t change
Quiz
Physics 1B03summer - Lecture 7
Energy
Since we know the total energy of a SHM, we can calculate the or displacement velocity at any point in time:
ETot = ½ kA2 = K+U = ½ mv2 + ½ kx2
So, if x=0, all E is in kinetic, and v is at max
if x=A, all E is in potential, and v is zero
Physics 1B03summer - Lecture 7
A 100g block is 5cm from the equilibrium position moving at 1.5m/s. The angular frequency is 2 rad/s.
a) What is the total energy of the system ?
b) What is the amplitude of the oscillations ?
Example 1
Physics 1B03summer - Lecture 7
Example 2
A 500g block on a spring is pulled 20cm and released.The motion has a period of 0.8s.
What is the velocity when the block is 15.4cm from the equilibrium ?
Physics 1B03summer - Lecture 7
A 1.0kg block is attached to a spring with k=16N/m. While the block is at rest, a student hits it with a hammer and almost instantaneously gives it a speed of 40cm/s.
a) what is the amplitude of the subsequent oscillations ?
b) what is the block’s speed at the point where x=A/2 ?
Example 3
Physics 1B03summer - Lecture 7
10 min rest
Physics 1B03summer - Lecture 7
SHM and Circular Motion
)sin(sin
)cos(cos
0
0
tAAy
tAAx
A
Uniform circular motion about in the xy-plane, radius A, angular velocity :
(t) = 0 + t
(similar to, x=xo+vt)
and so
Hence, a particle moving in one dimension can be expressedas an ‘imaginary’ particle moving in 2D (circle), or vice versa -the ‘projection’ of circular motion can be viewed as 1D motion.
Physics 1B03summer - Lecture 7
)sin(sin
)cos(cos
tAAy
tAAx
o
o
Compare with our expression for 1-D SHM.
Result:
SHM is the 1-D projection of uniform circular motion.
)cos( tAx
Physics 1B03summer - Lecture 7
Phase Constant, θo
A
For circular motion,the phase constantis just the angle atwhich the motion started.
Physics 1B03summer - Lecture 7
Example
An object is moving in circular motion with an angular frequency of 3π rad/s, and starts with an initial angle of π/6. If the amplitude is 2.0m, what is the objects angular position at t=3sec ?
What are the x and y values of the position at this time ?
Physics 1B03summer - Lecture 7
Simple Pendulum
Gravity is the “restoring force” taking the place of the “spring” in our block/spring system.
Instead of x, measure the displacement as the arc length s along the circular path.
Write down the tangential component of F=ma:
sinmg
L
θ T
sin
But
)sin(
2
2
2
2
L
g
dt
d
Ls
mgmadt
sdm t
mg sin θ
Restoring force
s
mg
Physics 1B03summer - Lecture 7
sin
2
2
L
g
dt
dSimple pendulum:
xdt
xd 22
2
SHM:
The pendulum is not a simple harmonic oscillator!
Lg
Lg
dtd sin 2
2
sinHowever, take small oscillations: (radians) if is small.
Then
22
2
dt
d
Compare:
Physics 1B03summer - Lecture 7
Simple Pendulum
L
g
dt
d
2
2
Using sin(θ)~θ for small angles, we have thefollowing equation of motion:
With:Lg
Hence:2
2
2 4gTg
L
Application - measuring height - finding variations in g → underground resources
or: 2
22 4
T
LLg
-------------------------------------------------------------------------
Physics 1B03summer - Lecture 7
This looks like
Lg
dtd 2
2
For small :
xdt
xd 22
2
, with angle instead of x.
So, the position is given by )cos()( o tt
amplitude
phase constant
(2/ period)
Physics 1B03summer - Lecture 7
a) longer.b) same.c) shorter.
Quiz:
A geologist is camped on top of a large deposit of nickel ore, in a location where the gravitational field is 0.01% stronger than normal. The period of his pendulum will be
(by how much, in percent?)
Physics 1B03summer - Lecture 7
Application
Pendulum clocks (“grandfather clocks”) often have a swinging arm with an adjustable weight. Suppose the arm oscillates with T=1.05sec and you want to adjust it to 1.00sec. Which way do you move the weight?
?
Physics 1B03summer - Lecture 7
Quiz
A simple pendulum hangs from the ceiling of an elevator. If the elevator accelerates upwards, the period of the pendulum:
a) Gets shorterb) Gets largerc) Stays the same
Question: What happens to the period of a simple pendulum if the mass m is doubled?
Physics 1B03summer - Lecture 7
A physicist wants to know the height of a building. He notes that a long pendulum that extends from the ceiling almost to the floor has a period of 12 s. How tall is the building?
Example
Physics 1B03summer - Lecture 7
Compare Springs and Pendulum
xm
k
dt
xd
2
2
kxmaF
Newton’s 2nd Law:
mk
L
g
dt
d
2
2
Lg
singmmaF
Physics 1B03summer - Lecture 7
t
x
x
t
SHM: x(t) = A cos ωt Motion continues indefinitely. Only conservative forces act, so the mechanical energy is constant.
Damped oscillator: dissipative forces (friction, air resistance, etc.) remove energy from the oscillator, and the amplitude decreases with time.
SHM and Damping – EXTRA !!!
Physics 1B03summer - Lecture 7
)cos()( 2
tAetxt
m
b
For weak damping (small b), the solution is:
f = bv where b is a constant damping coefficient
x
t
A damped oscillator has external nonconservativeforce(s) acting on the system. A common example is a force that is proportional to the velocity.
eg: green water
A e-(b/2m)t
2
2
dt
xdm
dt
dxbkx F=ma give:
Physics 1B03summer - Lecture 7
10 min rest
Physics 1B03summer - Lecture 7
Wave Motion Wave Motion – Chapter 20– Chapter 20
•Qualitative properties of wave motion•Mathematical description of waves in 1-D•Sinusoidal waves
Physics 1B03summer - Lecture 7
A wave is a moving pattern. For example, a wave on a stretched string:
The wave speed v is the speed of the pattern. No particles move at this wave speed – it is the speed of the wave. However, the wave does carry energy and momentum.
v
Δx = v Δt
time t
t +Δt
Physics 1B03summer - Lecture 7
The wave moves this way
The particles move up and down
If the particle motion is perpendicular to the direction the wave travels, the wave is called a “transverse wave”.
Transverse waves
Examples: Waves on a string; waves on water; light & other electromagnetic waves; some sound waves in solids (shear waves)
Physics 1B03summer - Lecture 7
The wave moves long distances parallel to the particle motions.
Longitudinal waves
Example: sound waves in fluids (air, water)
Even in longitudinal waves, the particle velocities are quite different from the wave velocity. The speed of the wave can be orders of magnitude larger than the particle speeds.
The particles move back and forth.
Physics 1B03summer - Lecture 7
QuizQuiz
wave motionA
B
C
Which particle is moving at the highest speed?
A) AB) BC) CD) all move with the same speed
Physics 1B03summer - Lecture 7
For these waves, the wave speed is determined entirely by the medium, and is the same for all sizes & shapes of waves.
T
v lengthmass/unit
tension
eg. stretched string:
(A familar example of a dispersive wave is an ordinary water wave in deep water. We will discuss only non-dispersive waves.)
Non-dispersive waves: the wave always keeps the same shape as it moves.
Physics 1B03summer - Lecture 7
Principle of SuperpositionPrinciple of Superposition
When two waves meet, the displacements add:
),(),(),( 21 txytxytxy observed
So, waves can pass through each other:
v
v
(for waves in a “linear medium”)
Physics 1B03summer - Lecture 7
Quiz: Quiz: “equal and opposite” waves“equal and opposite” waves
v
v
Sketch the particle velocities at the instant the string is completely straight.
v
v
Physics 1B03summer - Lecture 7
ReflectionsReflections
Waves (partially) reflect from any boundaryin the medium:
1) “Soft” boundary:
Reflection is upright
light string, or free end
Physics 1B03summer - Lecture 7
Reflection is inverted
ReflectionsReflections
2) “Hard” boundary:
heavy rope, or fixed end.
Physics 1B03summer - Lecture 7
QuizQuiz
A) y= f(xvt) B) y= f(x+vt) C) y= f( xvt) D) y= f( x+vt)
A wave show can be described by the equation y=f(xvt), as it travels along the x axis. It reflects from a fixed end at the origin. The reflected wave is described by:
Physics 1B03summer - Lecture 7
The math: Suppose the shape of the wave at t = 0 is given by some function y = f(x).
y y = f (x)v
at time t :y = f (x - vt)
y
vt
at time t = 0:
Note: y = y(x,t), a function of two variables; f is a function of one variable
Physics 1B03summer - Lecture 7
Non-dispersive waves:
y (x,t) = f (x ± vt)
+ sign: wave travels towards –x sign: wave travels towards +x
f is any (smooth) function of one variable.
eg. f(x) = A sin (kx)