Physics 201: Lecture 29, Pg 1 Lecture 29 Goals Goals Describe oscillatory motion in a simple...
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Transcript of Physics 201: Lecture 29, Pg 1 Lecture 29 Goals Goals Describe oscillatory motion in a simple...
Physics 201: Lecture 29, Pg 1
Lecture 29
• GoalsGoals Describe oscillatory motion in a simple pendulum Describe oscillatory motion with torques Introduce damping in SHM Discuss resonance
Physics 201: Lecture 29, Pg 2
Final Exam Details
Sunday, May 13th 10:05am-12:05pm in 125 Ag Hall & quiet room
Format: Closed book Up to 4 8½x1 sheets, hand written only Approximately 50% from Chapters 13-15 and 50% 1-12 Bring a calculator
Special needs/ conflicts:
All requests for alternative test arrangements should be made by Thursday May10th (except for medical emergency)
Physics 201: Lecture 29, Pg 3
Mechanical Energy of the Spring-Mass System
Kinetic energy is always
K = ½ mv2 = ½ m(A)2 sin2(t+)
Potential energy of a spring is,
U = ½ k x2 = ½ k A2 cos2(t + )
And 2 = k / m or k = m 2
K + U = constant
x(t) = A cos( t + )
v(t) = -A sin( t + )
a(t) = -2A cos( t + )
Physics 201: Lecture 29, Pg 4
SHM is a close as Lake Mendota…
So can you estimate the characteristic frequency for a bobbing in the water?
If you have equilibrium and there is a linear restoring force, then yes with = (k / m)½
F mgB
y0
mgFF By 0 equlibriumAt
gAymgF wB 0
Physics 201: Lecture 29, Pg 5
SHM is a close as Lake Mendota…
Deeper than y0 means
and a net force of
A linear restoring force with k = wAg
and boat mass m = Ay0 w so = (g / y0)½
Lighter boats bob more quickly than heavy ones
(if the same size)
FB mg
y
AygF wB gyyAF w )( 0Net
Physics 201: Lecture 29, Pg 7
The shaker cart You stand inside a small cart attached to a heavy-duty
spring, the spring is compressed and released, and you shake back and forth, attempting to maintain your balance. Note that there is also a sandbag in the cart with you.
At the instant you pass through the equilibrium position of the spring, you drop the sandbag out of the cart onto the ground.
What effect does jettisoning the sandbag at the equilibrium position have on the amplitude of your oscillation?
A. It increases the amplitude.B. It decreases the amplitude.C. It has no effect on the amplitude.
Hint: At equilibrium, both the cart and the bag are moving at their maximum speed.
Physics 201: Lecture 29, Pg 8
The shaker cart
Instead of dropping the sandbag as you pass through equilibrium, you decide to drop the sandbag when the cart is at its maximum distance from equilibrium.
What effect does jettisoning the sandbag at the cart’s maximum distance from equilibrium have on the amplitude of your oscillation?
A. It increases the amplitude.
B. It decreases the amplitude.
C. It has no effect on the amplitude.
Hint: At maximum displacement there is no kinetic energy.
Physics 201: Lecture 29, Pg 9
The shaker cart
What effect does jettisoning the sandbag at the cart’s maximum displacement from equilibrium have on the maximum speed of the cart?
A. It increases the maximum speed.
B. It decreases the maximum speed.
C. It has no effect on the maximum speed.
Hint: At maximum displacement there is no kinetic energy.
Physics 201: Lecture 29, Pg 10
The Pendulum (using torque)
A pendulum is made by suspending a mass m at the end of a string of length L. Find the frequency of oscillation for small displacements.
z = Iz = -mg sin() L
z ≈ mL2z ≈ -mg L
L (d2 /dt2) = -g
compare to max = -kx
d2 /dt2 = (-g/L)
with (t)= 0 cos( t + )
and =(g/L)½
z
L
mg
y
x
T
L sin
Physics 201: Lecture 29, Pg 11
The Pendulum
A pendulum is made by suspending a mass m at the end of a string of length L. Find the frequency of oscillation for small displacements.
If small then sin()
0° tan 0.00 = sin 0.00 = 0.00
5° tan 0.09 = sin 0.09 = 0.09
10° tan 0.17 = sin 0.17 = 0.17
15° tan 0.26 = 0.27
sin 0.26 = 0.26
L
m
mg
z
y
x
T
Physics 201: Lecture 29, Pg 18
Exercise Simple Harmonic Motion
You are sitting on a swing. A friend gives you a small push and you start swinging back & forth with period T1.
Suppose you were standing on the swing rather than sitting. When given a small push you start swinging back & forth with period T2.
Which of the following is true recalling that = (g/L)½ (A) T1 = T2
(B) T1 > T2
(C) T1 < T2
Physics 201: Lecture 29, Pg 19
A Rod Pendulum
A pendulum is made by suspending a thin rod of length L and mass M at one end. Find the frequency of oscillation for small displacements.
z = I = -| r x F | = (L/2) mg sin()
Irod at end = mL2/3
- mL2/3 L/2 mg -1/3 L d2/dt2 = ½ g
Lmg
z
xCM
T
L
g
2
3
Physics 201: Lecture 29, Pg 21
Torsion Pendulum Consider an object suspended by a wire
attached at its CM. The wire defines the rotation axis, and the moment of inertia I about this axis is known.
The wire acts like a “rotational spring”. When the object is rotated, the wire
is twisted. This produces a torque that opposes the rotation.
Torque is proportional to the angular displacement: = - where is the torsion constant
= (/I)½
I
wire
Physics 201: Lecture 29, Pg 22
Exercise Period All of the following torsional pendulum bobs have
the same mass and radius with = (/I)½
Which pendulum rotates the slowest (i.e. has the
longest period) if the wires are identical?
RRRR
(A) (B) (C) (D)
Physics 201: Lecture 29, Pg 23
What about Friction?A velocity dependent drag force (A model)
2
2
dtxd
mdtdx
bkx
We can guess at a new solution.
)(cos expA )( )(2
ttxm
bt
With,
22
2
22
mb
mb
mk
o
02
2
xm
k
dt
dx
m
b
dt
xd
and now 02 ≡ k / m
Note
Physics 201: Lecture 29, Pg 24
What about Friction?
A damped exponential
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
t
A)(cos
2expA )( )( tt
mbtx mbo 2/if
Physics 201: Lecture 29, Pg 25
Variations in the damping
Small damping time constant (m/b)
Low friction coefficient, b << 2m
Moderate damping time constant (m/b)
Moderate friction coefficient (b < 2m)
Physics 201: Lecture 29, Pg 26
Damped Simple Harmonic Motion
A downward shift in the angular frequency There are three mathematically distinct regimes
22 )2/( mbo
mbo 2/ mbo 2/
underdamped critically damped overdamped
mbo 2/
Physics 201: Lecture 29, Pg 28
Driven SHM with Resistance Apply a sinusoidal force, F0 cos (t), and now consider what A and b do,
2220
2
0
)()(
/
mb
mFA
b/m small
b/m middling
b large
tm
Fx
m
k
dt
dx
m
b
dt
xd cos 2
2
Not Zero!!!
stea
dy s
tate
am
plitu
de
Physics 201: Lecture 29, Pg 29
For Thursday
Review for final!