Physics 211: Lecture 25, Pg 1 Physics 211: Lecture 25 Physics 211: Lecture 25 Today’s Agenda Today’s Agenda Recap of last lecture Using “initial conditions” to solve problems The general physical pendulum The torsion pendulum Energy in SHM Atomic Vibrations Problem: Vertical Spring Problem: Transport Tunnel SHM Review
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Physics 211: Lecture 25, Pg 1
Physics 211: Lecture 25Physics 211: Lecture 25
Today’s AgendaToday’s Agenda Recap of last lecture Using “initial conditions” to solve problems The general physical pendulum The torsion pendulum Energy in SHM
Atomic Vibrations Problem: Vertical Spring Problem: Transport Tunnel SHM Review
Physics 211: Lecture 25, Pg 2
SHM and SpringsSHM and Springs
k
s
m
0
k
m
s
0
d sdt
s2
22
km
Force:
Solution:s = A cos(t + )
Physics 211: Lecture 25, Pg 3
Velocity and AccelerationVelocity and Acceleration
k
x
m
0
Position: x(t) = A cos(t + ) Velocity: v(t) = -A sin(t + )Acceleration: a(t) = -2A cos(t + )
A mass oscillates up & down on a spring. Its position as a function of time is shown below. At which of the points shown does the mass have positive velocity and negative acceleration?
The slope of y(t) tells us the sign of the velocity since
t
y(t)
(a)
(b)
(c)
v dydt
y(t) and a(t) have the opposite sign since a(t) = -2 y(t)
a < 0v < 0
a > 0v > 0
a < 0v > 0
The answer is (c).
Physics 211: Lecture 25, Pg 6
ExampleExample
A mass m = 2 kg on a spring oscillates with amplitude A = 10 cm. At t = 0 its speed is maximum, and is v = +2 m/s. What is the angular frequency of oscillation ? What is the spring constant k?
k
x
m
= 1MAX s20cm10
sm2A
v
km
Also: k = m2
So k = (2 kg) x (20 s -1) 2 = 800 kg/s2 = 800 N/m
vMAX = A
Physics 211: Lecture 25, Pg 7
Initial ConditionsInitial Conditions
k
x
m
0
Use “initial conditions” to determine phase !
Suppose we are told x(0) = 0 , and x is initially increasing (i.e. v(0) = positive):
x(t) = A cos(t + ) v(t) = -A sin(t + )a(t) = -2A cos(t + )
sincos
x(0) = 0 = A cos() = /2 or -/2 0 < v(0) = -A sin() < 0
= -/2So
Physics 211: Lecture 25, Pg 8
Initial Conditions...Initial Conditions...
k
x
m
0
x(t) = A cos(t - /2 ) v(t) = -A sin(t - /2 )a(t) = -2A cos(t - /2 )
A mass hanging from a vertical spring is lifted a distance d above equilibrium and released at t = 0. Which of the following describes its velocity and acceleration as a function of time?
Since we start with the maximum possibledisplacement at t = 0 we know that:
y = d cos(t)
t = 0
v dydt
d sin t v sin tmax
a dvdt
d cos t a cos tmax 2
Physics 211: Lecture 25, Pg 11
Review of Simple PendulumReview of Simple Pendulum
Using = Iand sin for small
L
dm
mg
z
mgL mL ddt
22
2
I We found
ddt
2
22
Lgwhere
Which has SHM solution = 0 cos(t + )
Physics 211: Lecture 25, Pg 12
Review of Rod PendulumReview of Rod Pendulum
Using = Iand sin for small
Ldmg
z
L/2
xCM
mg L mL ddt2
13
22
2
I
ddt
2
22 3
2gL
where
We found
Which has SHM solution = 0 cos(t + )
Physics 211: Lecture 25, Pg 13
General Physical PendulumGeneral Physical Pendulum
Suppose we have some arbitrarily shaped solid of mass M hung on a fixed axis, and that we know where the CM is located and what the moment of inertia I about the axis is.
The torque about the rotation (z) axis for small is (sin )
SHM and quadratic potentials...SHM and quadratic potentials...
U
x
x0
U
x
U(x) = U (x0) x 2
Let k = U (x0)
Then:
U(x) = k x 2
21
21
SHM potential!!
Physics 211: Lecture 25, Pg 22
Problem: Vertical SpringProblem: Vertical Spring
A mass m = 102 g is hung from a vertical spring. The equilibrium position is at y = 0. The mass is then pulled down a distance d = 10 cm from equilibrium and released at t = 0. The measured period of oscillation is T = 0.8 s. What is the spring constant k? Write down the equations for the position,
velocity, and acceleration of the mass as functions of time.
What is the maximum velocity? What is the maximum acceleration?
A straight tunnel is dug from Urbana through the center of the Earth and out the other side. A physics 211 student jumps into the hole at noon. What time does she get back to Urbana?
Physics 211: Lecture 25, Pg 27
Transport Tunnel...Transport Tunnel...
R
RE
F R GmMRG
R 2
where MR is themass inside radius R
MR
FG
F RF R
MR
RM
G
G E
R E
E 2
2
but 3R RM
F RF R
RR
RR
RR
G
G E
E
E E
3
2
2
3
Physics 211: Lecture 25, Pg 28
Transport Tunnel...Transport Tunnel...
R
RE MR
FG
EEG
G
RR
RFRF
F mg RR
kRGE
k mgRE
Like a mass ona spring with
mg)R(F EG
Physics 211: Lecture 25, Pg 29
Transport Tunnel...Transport Tunnel...
R
RE MR
FG
k mgRE
Like a mass ona spring with
km
gRE
So:
plug in g = 9.81 m/s2
and RE = 6.38 x 106 m
get = .00124 s-1
and so T = = 5067 s
84 min2
Physics 211: Lecture 25, Pg 30
Transport Tunnel...Transport Tunnel...
So she gets back to Urbana 84 minutes later, at 1:24 p.m.
Physics 211: Lecture 25, Pg 31
Transport Tunnel...Transport Tunnel...
Strange but true: Strange but true: The period of oscillation does not require that the tunnel be straight through the middle!! Any straight tunnel gives the same answer, as long as it is frictionless and the density of the Earth is constant.
Physics 211: Lecture 25, Pg 32
Transport Tunnel...Transport Tunnel...
Another strange but true Another strange but true fact: fact: An object orbiting the earth near the surface will have a period of the same length as that of the transport tunnel.
An object with mass M, radius R and moment of inertia I = MR2 rolls without slipping down an incline which makes an angle with respect to the horizontal. 13. Which of the following gives the correct expression for the translational acceleration, a , of the object?
a. sin1
ga
b. sina g
c. sin1
a g
d. 1 sina g
e. sin2
ga
R
a g
Physics 211: Lecture 25, Pg 35
Physics 211: Lecture 25, Pg 36
Physics 211: Lecture 25, Pg 37
Physics 211: Lecture 25, Pg 38
Recap of today’s lectureRecap of today’s lecture
Recap of last lecture Using “initial conditions” to solve problems
(Text: 14-1) The general physical pendulum (Text: 14-3) The torsion pendulum Energy in SHM (Text: 14-2)
Atomic Vibrations Problem: Vertical Spring (Text: 14-3) Problem: Transport Tunnel SHM Review
Look at textbook problems Look at textbook problems Chapter 14: # 61, 63, 67, 68, 74, 122