Simple Harmonic Motion

Physics 202Professor Lee

CarknerLecture 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 () -- = 2f = 2/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(t + )

Amplitude (xm) --

Phase angle () --

Remember that (t+) is in radians

SHM in Action Consider SHM with =0:

x = xmcos(t) Remember =2/T

x=xm

x=-xm

x=xm

SHM Monster

RestMin Max

10m

Phase

The value of relative to 2 indicates the offset as a function of one period It is phase shifted by 1/2 period

Velocity

v(t)=-xmsin(t + )

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)=-xmcos(t + )

Making a substitution yields:a(t)=-2x(t)

x, v and a Consider SMH with =0:

x = xmcos(t)v = -xmsin(t) = -vmsin(t)

a = -xmcos(t) = -amcos(t)

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=-m2x

F=-kx Where k=m2 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=m2

mk

ω

km

2πT

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) Higherb) Lowerc) 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) Higherb) Lowerc) 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) Higherb) Lowerc) 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) Both the sameb) Both higherc) Both lowerd) Flow rate is same, velocity is highere) 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) Higherb) Lowerc) The samed) 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) Higherb) Lowerc) The samed) Can’t tell