p150c11:1

Chapter 11: Harmonic MotionElastic Potential Energy

energy stored in a stretched/compressed spring

Force: Hooke’s Law F = kx

F = 0

F = k x/2

F = k x

x

unstretched

half stretched

stretched

2

2

21

21

2

springin storedenergy string stretching donework

kxPE

kxxx

kxFavg

p150c11:2

A horizontal spring had a force constant of 90 N/m. A mass of 1.5 kg is attached free end. The spring is then compressed by 50 cm, and then released. What is the speed of the mass when it returns to the equilibrium position?

p150c11:3

Simple Harmonic Motion: oscillations

Period T = time for one complete oscillation

frequency f = number of oscillations per time

usually number per second (1 cycle/sec = 1 Hertz = 1Hz)

f = 1/T

for a restoring force (equilibrium) + inertia

Fr = kx plus F = ma

mk

f

km

T

21

2

p150c11:4

Position, speed and acceleration in simple harmonic motion

x

t

maximum speed at x = 0

Amplitude A = xmax

v = 0 when x = xmax

x = A cos 2ft = A cos t

fAv

xAfv

mk

f

mvkxkA

2and

2

2with

21

21

21

KE+ PE =Energy Total

max

22

222

p150c11:5

Example: An object undergoes SHM with a frequency of 20 Hz and a maximum speed of 2.5 m/s. What is the amplitude of the motion? What is the object’s displacement when its speed is 1.5 m/s?

p150c11:6

The Simple Pendulum

mass m on a string of length L

x

s

mg

T

Fnet

LL

gL

T

Lg

mk

f

Lmg

kxL

mgF

mgF

Lx

net

net

2

21""

21

""

SHO =>force restoring

)geometry from(

Example: How long should a pendulum be in order to have a period of 1.0 s?