9.1 Simple Harmonic Motion Flashcards
What is simple harmonic motion?
The acceleration and the net force are PROPORTIONAL TO and in the OPPOSITE DIRECTION to the displacement from equilibrium.
What is the formula for simple harmonic motion?
a = -w^2x
w is constant called the angular frequency.
What does SHM consist of?
Periodic oscillations with a period that is independent of the amplitude.
What is the period?
T = 2pi/w
What is the phase?
An angle added to wt in the formulas for x,v and a which indicates their values at t=0.
What is the formula for potential energy?
Ep = 1/2mw^2x^2
What is the formula for kinetic energy?
Ek = 1/2mw^2(x0^2-x^2)
What are the two special cases of SHM?
A mass m at the end of a spring with spring constant k the period T = 2pi sqr rt(m/k)
A single pendulum of length l the period is
T = 2pi sqr rt(l/g)
What is important about the period expression for the pendulum?
The expression for the pendulum is true provided the amplitude is small. Note that the period of a pendulum does not depend on its mass.
Why is common choice of phase 0?
Since x = x0coswt
v = -wx0sinwt
a = -w^2xocoswt
The maximum speed is v max = wx0
the maximum acceleration is a max = w^2x0
Why is another common choice of phase
- pi/2
x = x0sinwt v = wx0coswt a = -w^2x0sinwt
What happens to the kinetic and potential energy in SHM?
They are constantly transferred into one another.
When does the maximum potential energy occur?
1/2mw^2x0^2 occurs at maximum displacement where the speed is 0. Therefore the kinetic energy is also zero at this point so the formula above is a measure of the total energy.
How is the kinetic energy at any displacement given?
Since ET = EK + EP at any point the keitnc energy at any displacement is given by
1/2mv^2 = 1/2mw^2x0^2 - 1/2mw^2x^2
so v = plus or minus w times sqr rt(x0^2 - x^2)
What happens as the oscillating body goes past the equilibrium point?
displacement is zero and so the velocity is equal to plus or minus wx0.
