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:

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

Send me an email at calebjasik@jasik.xyz if you want to.