September 2, 2018

PHYS 221.501 | First Class Notes

Characteristics of periodic motion

Amplitude is maximum magnitude of displacement from equilibrium Period, \(T\), is the time for one cycle Frequency, \(F\) is the number of cycles per unit of time = \(F = \frac{1}{T}\) Angular Frequency,

Test Your Understanding of Section 4.1 in the book

Simple harmonic motion:

When the restoring force is directly proportional to the displacement from equilibrium, the resulting motion is called simple harmonic motion, i.e., \(F_x = -kx\).

Simple harmonic motion can be considered a projection of uniform circular motion.

Characteristics of SHM For a body that is vibrating via an ideal spring: \(F_x = -kx = m a_x = x\omega^2\)

If you increase the mass of an object in SMH, the frequency will decrease.

\(\omega = \sqrt{\frac{k}{m}}\)

Displacement as a function of time in SHM

\(x = Acos(\omega t)\)

\(T = 2\pi\sqrt{\frac{m}{k}}\)

Graphs of displacement, velocity, and acceleration:

The displacement as a function of time for SHM with phase angle \(\phi\) is \(x = Acos(\omega t + \phi)\)

The velocity is \(\frac{dx}{dt} = v_x = -A\omega sin(\omega t + \phi)\)

The acceleration is \(\frac{d^2x}{dt^2} = a_x = -A\omega^2 cos(\omega t + \phi) = -\omega^2 x\)

The phase angle, \(\phi\) refers to where in the period that you begin the curve.

Total mechanical energy is conserved:

\(E = \frac{1}{2}mv_x^2 + \frac{1}{2}kx^2 = \frac{1}{2}kA^2 = constant\)