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Department of Physics and Applied Physics95.141, Spring 2014, Lecture 23
Lecture 23
Chapter 14
OscillationsSimple pendulum
28.04.2014Physics I
Course website:http://faculty.uml.edu/Andriy_Danylov/Teaching/PhysicsI
Lecture Capture: http://echo360.uml.edu/danylov2013/physics1spring.html
Department of Physics and Applied Physics95.141, Spring 2014, Lecture 23
Exam 3 Results
Average 50.5/100
79
1614
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2623
26
21
35
2018
13
17
40
05
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frequency
grade
total grade distribution
grade distribution section average201 65.2381202 45.7203 57.0435204 48.6471205 50.2381206 53.8947207 64.2273208 45.4545209 50.6842210 54.1111211 44.8571212 42.375213 41.6667214 41.5556215 47
Department of Physics and Applied Physics95.141, Spring 2014, Lecture 23
Chapter 14
Simple Harmonic Motion Energy in the Simple Harmonic Oscillator The Simple Pendulum
Outline
Department of Physics and Applied Physics95.141, Spring 2014, Lecture 23
Simple Harmonic Motion
km
f 1T
T 2
amplitudedisplacement
frequency
phase angle
angular frequency
This function describes a simple harmonic motion
)cos()( tAtx
period of oscillations, T
kmT 2
xdt
xd 22
2
A Differential Equation
wherex
0x x
Department of Physics and Applied Physics95.141, Spring 2014, Lecture 23
2
2
dtxda
If we know x(t), we can calculate v(t) and a(t)
dtdxv
)cos( tAx
The velocity, v(t), and acceleration, a(t), for simple harmonic motion can be found by differentiating the displacement:
Av max
Aa 2max
)sin( tA
)cos(2 tA x2
Department of Physics and Applied Physics95.141, Spring 2014, Lecture 23
The Phase Angle: the effect of φ (I)
Recall calculus.
Let us start witha function,
We can move it upby adding a constant to the y-value:
2xy cxy 2
You can move it left or right by adding a constant to the x-value
2cxy
Department of Physics and Applied Physics95.141, Spring 2014, Lecture 23
What does the Phase Angle do? The effect of φ
t
)cos( tAx
)cos( tAx0φ =0
t
t
t
t)cos( tAx
φ >0
φ <0
Positive phase moves the curve left by φ/
Negative phase moves the curve right by φ/
Value of phase, φ, doesn’t affect the shape of the curve x(t)
)(cos
tA
)(cos
tA
Department of Physics and Applied Physics95.141, Spring 2014, Lecture 23
Exa
mpl
e: S
HM
=2/3x(t)=(10m)cos(3.14t+2/3)
Two identical blocks oscillate on different horizontal springs. Which spring has the larger spring constant?
A) The red spring.B) The blue spring.
C) There’s not enough information to tell.
ConcepTest 1 Spring constant
mkf (1/2
,ff since bluered
bluered kk then
Department of Physics and Applied Physics95.141, Spring 2014, Lecture 23
Demonstrations (mass/spring system).http://stopwatch.onlineclock.net/
kmT 2
The period is independent of Amplitude!
A spring/mass system can be used in space to measure mass of an object
Department of Physics and Applied Physics95.141, Spring 2014, Lecture 23
Energy in Simple Harmonic MotionPotential energy stored in spring
Kinetic energy of mass
U 12 kx2
It is a constant !
K 12 mv2
U 12 kA2 cos2(t ) K 1
2 kA2 sin2 (t )
UKESo
Total mechanical energy is conserved
dtdxv
Energy
x+A-A
K
U
E
)cos( tAx )sin( tA
221 kA
)](cos)([sin 22221 ttkA
UKE
Department of Physics and Applied Physics95.141, Spring 2014, Lecture 23
Oscillations. Summary
Ax 0xx
Ax k m
0v 0vAv max
0aAa 2max Aa 2
max
221 kAUE 2
max21 mvKE 2
21 kAUE
)cos( tAx)sin( tAv
)cos(2 tAa km
T 2
maxFF 0F maxFF
Department of Physics and Applied Physics95.141, Spring 2014, Lecture 23
Thank youSee you on Wednesday