Physics 101- Introduction to Mechanics
|April 24- Simple Harmonic Oscillators|
|What do the following
have in common?
Each can (at least in some way) be described by the
same basic physics/ math- simple harmonic oscillators.
Consider an ideal spring
For static situations (a hanging mass/ spring), you have a balance of forces so, kx= mg. (yawn)
Letís look at something new and different- dynamic/
oscillating situations. (For now let's consider a horizontal spring,
where the mass is on a frictionless tabletop.)
Plug our force expression into F=ma where you use
the definition of acceleration to get a differential equation:
What is the solution? What is x(t)? Well,
the answer is given as
where A is the amplitude (meter), w is the angular frequency (radians/ sec) (notice that w is NOT angular velocity, this is a new variable), and f is the phase constant (radians).
The angular frequency describes how fast
(or how frequently) an object is oscillating. The phase constant
is important since it helps us to describe where the object is at t= 0,
and consequentially at all times. Without f,
you wouldn't know if your object started at the top or bottom of its motion
(or somewhere else).
A glider on an air track is connected by a spring to the end of the air track. If it is pulled 3.5cm in the +x direction away from its equilibrium point and then released from rest at t=0, what is the phase constant f?
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