Transcript A Brief History of Planetary Science

Simple Harmonic Motion
Physics 202
Professor Lee Carkner
Lecture 3
PAL #2 Archimedes
Weigh object =
Weigh bowl =
Put cup full of water in bowl, insert object, remove
cup
Weigh bowl of water = mbowl2
Volume of object =
Density of object = mobj / Vobj
Biggest source of error is getting the right amount
of water into the cup

class answers range from 3000-15000 kg/m3
Would not work for floating object unless you held
it underwater
 Hard to estimate fraction submerged
Simple Harmonic Motion

e.g. a mass on a spring
Frequency (f) -Unit=hertz (Hz) = 1 oscillation per second = s-1
Period (T) -T=1/f
Angular frequency (w) -- w = 2pf = 2p/T
Unit =
We use angular frequency because the
motion cycles
Equation of Motion

The displacement from the origin of a particle
undergoing simple harmonic motion is:
x(t) = xmcos(wt + f)
Amplitude (xm) --
Phase angle (f) -Remember that (wt+f) is in radians
SHM in Action
Consider SHM with f=0:
x = xmcos(wt)
Remember w=2p/T

x=xm

x=-xm

x=xm
Min
Rest
Max
10m
SHM Monster

Phase
The value of f relative to 2p indicates the
offset as a function of one period

It is phase shifted by 1/2 period
Velocity

v(t)=-wxmsin(wt + f)

Since the particle moves from +xm to -xm the
velocity must be negative (and then positive in the
other direction)

High frequency (many cycles per second) means
larger velocity
Acceleration

a(t)=-w2xmcos(wt + f)

Making a substitution yields:
a(t)=-w2x(t)
x, v and a
 Consider SMH with f=0:
x = xmcos(wt)
v = -wxmsin(wt) = -vmsin(wt)
a = -w2xmcos(wt) = -amcos(wt)

 Mass is momentarily at rest, but
being pulled hard in the other
direction

 Mass coasts through the middle at
high speed
Force

But, F=ma so,
F=-mw2x

F=-kx
Where k=mw2 is the spring constant


Simple harmonic motion is motion where
force is proportional to displacement but
opposite in sign
Why is the sign negative?
Linear Oscillator
A simple 1-dimensional SHM system is
called a linear oscillator

In such a system, k=mw2

k
ω
m
m
T2π
k
Next Time
Read: 15.4, 15.6, 15.8, 15.9
Homework: Ch 15, P: 9, 28, 46, 78, 109
Consider a small ball on a toy ship
floating in a tank of water. If you
remove the ball from the ship and place
it on the table the water level will be,
a) Higher
b) Lower
c) The same
Consider a small ball on a toy ship
floating in a tank of water. If you
remove the ball from the ship and place
it in the tank and it floats, the water
level will be,
a) Higher
b) Lower
c) The same
Consider a small ball on a toy ship
floating in a tank of water. If you
remove the ball from the ship and place
it in the tank and it sinks to the bottom,
the water level will be,
a) Higher
b) Lower
c) The same
Water flows through a horizontal pipe
from point A to point B. If the pipe is
narrower at B than at A the flow rate
and velocity at point B compared to
point A are,
a)
b)
c)
d)
e)
Both the same
Both higher
Both lower
Flow rate is same, velocity is higher
Velocity is same, flow rate is higher
Water flows uphill through a pipe from
point A to point B. If the velocity is the
same at points A and B, the pressure at B
compared to A is,
a)
b)
c)
d)
Higher
Lower
The same
Can’t tell
Water flows through a horizontal pipe
from point A to point B. If the pressure
at point B is higher than that at point A,
the velocity at point B compared to A is
a)
b)
c)
d)
Higher
Lower
The same
Can’t tell