PHY 113 C Fall 2013 -- Lecture 15 110/17/2013

PHY 113 A General Physics I11 AM – 12:15 PM TR Olin 101

Plan for Lecture 15:

Chapter 15 – Simple harmonic motion

1. Object attached to a spring; pendulum

Displacement as a function of time

Kinetic and potential energy

2. Driven harmonic motion; resonance

PHY 113 C Fall 2013 -- Lecture 15 210/17/2013

PHY 113 C Fall 2013 -- Lecture 15 310/17/2013

From Webassign Assignment #13

A uniform beam of length L = 7.45 m and weight 4.95 102 N is carried by two workers, Sam and Joe, as shown in the figure below. Determine the force that each person exerts on the beam.

X

mg

FSam FJoe

d1

d2

0

0

0 0

12

mgddF

-mgFF

Joe

JoeSam

ii

ii τF

PHY 113 C Fall 2013 -- Lecture 15 410/17/2013

Illustration of “simple harmonic motion”.

PHY 113 C Fall 2013 -- Lecture 15 510/17/2013

Hooke’s lawFs = -k(x-x0)

x0 =0

Behavior of materials:

Young’s modulus

LL

EAF

FFLL

AFE

material

appliedmaterial

applied

/

/

Simple harmonic motion; mass and spring

PHY 113 C Fall 2013 -- Lecture 15 610/17/2013

Microscopic picture of material with springs representing bonds between atoms

Measurement of elastic response:

LL

EAF

FFLL

AFE

material

appliedmaterial

applied

/

/

k

PHY 113 C Fall 2013 -- Lecture 15 710/17/2013

F s (N

)

x

Us (

J)

x

Fs = -kxUs = ½ k x2

General potential energy curve:U

(J)

x

Us = ½ k (x-1)2

U(x)

1

2

2

x

dx

Udk

Potential energy associated with Hooke’s law:

Form of Hooke’s law for ideal system:

PHY 113 C Fall 2013 -- Lecture 15 810/17/2013

Motion associated with Hooke’s law forces

Newton’s second law:

F = -k x = m a

xmk

dt

xd

dt

xdmkxF

2

2

2

2

“second-order” linear differential equation

PHY 113 C Fall 2013 -- Lecture 15 910/17/2013

How to solve a second order linear differential equation:

Earlier example – constant force F0 acceleration a0

00

2

2

am

F

dt

xd

x(t) = x0 +v0t + ½ a0 t2

2 constants (initial values)

PHY 113 C Fall 2013 -- Lecture 15 1010/17/2013

Hooke’s law motion:

xmk

dt

xd

dt

xdmkxF

2

2

2

2

Forms of solution:

where:)ωsin()ωcos()(

)φωcos()(

tBtAtx

tAtx

mkω

2 constants (initial values)

PHY 113 C Fall 2013 -- Lecture 15 11

Verification:

Differential relations:

Therefore:

10/17/2013

)φω(sin ω)φωcos(

)φωcos(ω)φωsin(

tdt

td

tdt

td

)φω(cos ω)φωcos( 2

2

2

tA

dt

tAd

mk

thatprovidedxmk

dt

xd

satisfiestAtx

22

2

ω

)φω(cos)(

PHY 113 C Fall 2013 -- Lecture 15 1210/17/2013

) s(determine ω

)φω(cos)φω(cos ω)φωcos(

:equation thesatisfies guessthat Condition

2

22

2

m

k

tAm

ktA

dt

tAd

constantsunknown are and and where)cos()(

:form thehas )(for solution that Guess

:system spring-massfor law sNewton' :Recap

2

2

AtAtx

tx

xm

k

dt

xd

PHY 113 C Fall 2013 -- Lecture 15 1310/17/2013

iclicker exercise:Which of the following other possible guesses would provide a solution to Newton’s law for the mass-spring system:

above. theof one than More E.

)exp( D.

C.

)sin()cos( B.

sin A.

221

0

φωt

gttv

tBtA

)t(A

)sin()sin()cos()cos(os

)cos()sin()sin()cos(sin : thatNote

tAtA)t(cA

tAtA)t(A

xm

k

dt

xd

2

2

PHY 113 C Fall 2013 -- Lecture 15 1410/17/2013

iclicker questionHow do you decide whether to use

?sinor cos tAtxtAtx

A. Either can be a solution, depending on initial position and/or velocity

B. Pure guess

PHY 113 C Fall 2013 -- Lecture 15 1510/17/2013

tm

kxtxxA

)(m

kAt

dt

dx) (Axtx

tdt

dxxtx

A

cos)( and 0

sin0)0( cos)0(

0)0( and )0(

that know weSuppose

:conditions initial from and Finding

00

0

0

constantsunknown are and wherecos)(

:form thehas )(for solution that theknow now We

:system spring-massfor law sNewton'for solution Complete

2

2

Atm

kAtx

tx

xm

k

dt

xd

PHY 113 C Fall 2013 -- Lecture 15 1610/17/2013

tm

k

mk

vtx

mk

vA

)(m

kAvt

dt

dx) (Atx

vtdt

dxtx

A

sin/

)( /

and 2

sin)0( cos0)0(

)0( and 0)0(

that know weSuppose

:conditions initial from and Finding

00

0

0

constantsunknown are and wherecos)(

:form thehas )(for solution that theknow now We

:system spring-massfor law sNewton'for solution Complete

2

2

Atm

kAtx

tx

xm

k

dt

xd

PHY 113 C Fall 2013 -- Lecture 15 1710/17/2013

constantsunknown are and wherecos)(

:)(for Solution

:system spring-massfor law sNewton' :Review

2

2

Atm

kAtx

tx

xm

k

dt

xd

Tf

m

k

22

:period andfrequency toipRelationsh

frequency)(angular rad/s

:frequency sticcharacteri a has system spring-mass :Review

PHY 113 C Fall 2013 -- Lecture 15 1810/17/2013

iclicker exercise:

A certain mass m on a spring oscillates with a characteristic frequency of 2 cycles per second. Which of the following changes to the mass would increase the frequency to 4 cycles per second?

(a) 2m (b) 4m (c) m/2 (d) m/4

PHY 113 C Fall 2013 -- Lecture 15 1910/17/2013

Energy associated with simple harmonic motion

22222

22

)cos(2

1)sin(

2

1

)sin()(

2

1

2

1 :Energy

constants are and and where)cos()(

:ntdisplaceme of Form

tkAtAmE

tAdt

dxtv

kxmvE

Am

kωtAtx

2222

2

2

1)cos()sin(

2

1

But

kAttkAE

m

k

PHY 113 C Fall 2013 -- Lecture 15 2010/17/2013

Energy diagram:

x

U(x)=1/2 k x2

E=1/2 k A2

A

K

PHY 113 C Fall 2013 -- Lecture 15 2110/17/2013

Simple harmonic motion for a pendulum:

Q

L ) (since sinsin

αsinτ

22

2

2

2

mLILg

ImgL

dt

d

dt

d-IImgL

Approximation for small :Q

Lg

ω );φωcos()

:Solution

sin

2

2

tA(t

Lg

dt

d

PHY 113 C Fall 2013 -- Lecture 15 2210/17/2013

Pendulum example:

Q

L

Lg

ω );φωcos() tA(t

Suppose L=2m, what is the period of the pendulum?

s 84.2ω2π

T

T2π

rad/s 2135.22m

9.8m/sLg

ω2

PHY 113 C Fall 2013 -- Lecture 15 2310/17/2013

The notion of resonance:

Suppose F=-kx+F0 sin(Wt)

According to Newton:

)sin(

:)ous"inhomogene("equation alDifferenti

)sin(

02

2

2

2

0

tmF

xmk

dtxd

dtxd

mtFkx

)sin(ω

/)sin(

/

/)(

:Solution

220

20 t

mFt

mk

mFtx

PHY 113 C Fall 2013 -- Lecture 15 2410/17/2013

)sin(ω

/)sin(

/

/)(

220

20 t

mFt

mk

mFtx

2

2

0 )sin(dt

xdmtFkx

Physics of a “driven” harmonic oscillator:

“driving” frequency

“natural” frequency

w

220

ω

/

mF

(W=2rad/s)

(mag)

PHY 113 C Fall 2013 -- Lecture 15 2510/17/2013

Examples:

Suppose a mass m=0.2 kg is attached to a spring with k=1.81N/m and an oscillating driving force as shown above. Find the steady-state displacement x(t).

F(t)=1 N sin (3t)

m )3sin( 001m )3sin(32.0/81.1

2.0/1)sin(

/

/)(

220 ttt

mk

mFtx

Note: If k=1.90 N/m then:

m )3sin( 01m )3sin(32.0/90.1

2.0/1)sin(

/

/)(

220 ttt

mk

mFtx

PHY 113 C Fall 2013 -- Lecture 15 2610/17/2013

Tacoma Narrows bridge resonance Images from Wikipedia

Rebuilt bridge in Tacoma WA

PHY 113 C Fall 2013 -- Lecture 15 2710/17/2013

Bridge in July, 1940

PHY 113 C Fall 2013 -- Lecture 15 2810/17/2013

Collapse of bridge on Nov. 7, 1940

http://en.wikipedia.org/wiki/Tacoma_Narrows_Bridge_%281940%29

PHY 113 C Fall 2013 -- Lecture 15 2910/17/2013

Effects of gravity on spring motion

x=0

k

mgd

mgkd

0 :mequilibriuAt

tm

kAdtx

dxm

k

k

mgx

m

k

dt

dxd

gxm

k

dt

xd

dt

xdmmgkxF

cos)(

:Rewriting

2

2

2

2

2

2

PHY 113 C Fall 2013 -- Lecture 15 3010/17/2013

Example

Suppose a mass of 2 kg is attached to a spring with spring constant k=20 N/m. At t=0 the mass is displaced by 0.4 m and released from rest.

What is the subsequent displacement as a function of time?

What is the total energy?:

tmtx

sradsradm

k

1622.3cos4.0)(

/1622.3/2

20

JJ

kAkxmvE

6.14.0202

1

2

1

2

1

2

1

2

222

PHY 113 C Fall 2013 -- Lecture 15 3110/17/2013

Example

Suppose a mass of 2 kg is attached to a spring with spring constant k=20 N/m. The mass is attached to a device which supplies an oscillatory force of the form

What is the subsequent displacement as a function of time?

ttA

ttAtx

tmF

tAtx

tFkxdt

xdm

5sin06667.01623.3cos

5sin52/20

2/21623.3cos

sin/

cos :Solution

sin

:equation sNewton'

2

220

02

2

)5sin(2 tNF