Describe the displacement, velocity, and acceleration of an object exhibiting SHM

For an object exhibiting SHM, the displacement of that object measured from its equilibrium position can be represented by the equations

Minima, maxima, and zeros of displacement, velocity, and acceleration are features of harmonic motion.

Recognizing the positions or times at which the displacement, velocity, and acceleration for SHM have extrema or zeros can help in qualitatively describing the behavior of the motion.

The position as a function of time for an object exhibiting SHM is a solution of the second order differential equation derived from the application of Newton’s second law. \[\dfrac{d^2 x}{dt^2} = -\omega^2 x\]

Characteristics of SHM, such as velocity and acceleration, can be determined by or derived from the equation \[x = A \cos{(\omega t + \phi)}\]

The acceleration of an object exhibiting SHM is related to the object’s angular frequency and position. \[a = -\omega^2 x\]

It can be shown that the maximum velocity and acceleration of an object exhibiting SHM are related to the angular frequency of the object’s motion. \[v_{max} = A \omega\] \[a_{max} = A \omega^2\]

In the presence of a sinusoidal external force, a system may exhibit resonance.

Resonance occurs when an external force is exerted at the natural frequency of an oscillating system

Resonance increases the amplitude of oscillating motion.

The natural frequency of a system is the frequency at which the system will oscillate when it is displaced from its equilibrium position.

Changing the amplitude of a system exhibiting SHM will not change its period.

Properties of SHM can be determined and anayzed using graphical representations.