simple harmonic motion - physics with dr....
TRANSCRIPT
Frequency
Frequency = f
= number of oscillations in 1
second
Unit = 1/s = hz
Period = T
= time for 1 oscillation/cycle
Unit = s
Tf
fT
1
1
Mathematical description
A = amplitude = center to max height or trough ω = angular frequency φ = phase angle = where (in radians) the wave started
)cos()( tAtx
fT
12
2f
x(t), v(t), a(t)
Position is given by the cosine wave. Differentiate for velocity and acceleration. Note: a = -ω2x (characteristic of SHM)
)cos(
)sin(
)cos()(
2
tAdt
dva
tAdt
dxv
tAtx
Identify SHM
constant) (positive
constant) (positive
xa
constant) (positive
constant) (positive
xF
Motion obeys these relationships:
Forces look like this:
SHM problem
A block of mass 680 g is fastened to a spring of spring constant 65 N/m. The block is pulled 11 cm from its equilibrium position at x=0 on a frictionless surface and released from rest at t=0.
a) What force does the spring exert on the block just before the block is released?
NmmNkxF 2.7)11.0)(/65(
… SHM problem
b) What are the angular frequency, the frequency, and the period of the resulting oscillation?
k = 65 N/m m = 680 g x = 11 cm
rad/s 78.9
kg 68.0
N/m 65
m
k
hz 56.12
rad/s 8.9
2
f
s 64.0hz 56.1
11
fT
… SHM problem
c) What is the amplitude of oscillation?
k = 65 N/m m = 680 g x = 11 cm
= 9.8 rad/s f = 1.6 hz T = 0.64 s
xmax = 11 cm
… SHM problem
d) What is the block’s maximum speed?
k = 65 N/m m = 680 g x = 11 cm
= 9.8 rad/s f = 1.6 hz T = 0.64 s
)sin(
)cos()(
tAdt
dxv
tAtx
m/s 1.1)m 11.0)(rad/s 8.9()( maxmax xv
Hint:
… SHM problem.
e) What is the magnitude of the block’s maximum acceleration?
k = 65 N/m m = 680 g x = 11 cm
= 9.8 rad/s f = 1.6 hz T = 0.64 s
)cos(
)cos()(
2
2
2
tAdt
xda
tAtx
22
max
2
max m/s 11)m 11.0()rad/s 8.9()( xa
Hint:
Period of a pendulum
Simple pendulum:
Physical pendulum:
I = rotational inertia (moment of inertia)
about the pivot point
h = distance from pivot to center of mass
m = mass of physical pendulum
g
LT 2
mgh
IT 2
Superposition of waves
Waves add mathematically:
Wave 1:
Wave 2:
)cos()( 1111 tAtx
)cos()( 2222 tAtx
)cos()cos()()( 22211121 tAtAtxtx
Superposition of Waves
Fourier transform: Play with waves of different frequencies: http://teacher2.smithtown.k12.ny.us/winters/Si
mulations.htm
Original time domain wave
Frequency domain wave