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

1715

18

2623

26

21

35

2018

13

17

40

05

1015202530354045

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