15.2
Characteristics of Simple Harmonic Motion
The key characteristic of the simple harmonic motion is that the acceleration of the system and, therefore, the net force are proportional to the disp…
Consider a ruler with one end fixed to a tabletop and the other end with a mass attached. When the ruler-mass system is pulled up from the free end and released, it executes simple harmonic motion. If the displacement is plotted with respect to time, this results in a sinusoidal waveform.
Since the oscillatory motion begins when the mass is displaced from the equilibrium position, the equation for displacement can be expressed as the amplitude, A, multiplied by the cosine function of the time-dependent angular frequency and initial phase.
Phase shift describes the difference between two similar waveforms in terms of the time interval and is measured in radians.
Recall that the first derivative of the displacement function with respect to time is velocity, and the second derivative is acceleration.
In simple harmonic motion, the velocity is a sine function that lags the displacement by phase pi by 2 and is maximum at the equilibrium, whereas the acceleration is a cosine function that lags by phase pi and is maximal at the positions of maximum displacement.
View the full transcript and gain access to JoVE Core videos
Q1: What does the displacement equation represent in simple harmonic motion?
The displacement equation expresses position as amplitude multiplied by the cosine function of time-dependent angular frequency and initial phase. When a ruler-mass system is pulled and released, this equation describes the sinusoidal waveform observed when plotting displacement versus time. The phase shift component measures the time interval difference between similar waveforms in radians.
Q2: How do velocity and acceleration relate to displacement in simple harmonic motion?
Velocity is the first derivative of displacement and lags by phase pi/2, reaching maximum at equilibrium. Acceleration is the second derivative and lags by phase pi, becoming maximal at maximum displacement positions. Both velocity and acceleration follow sine and cosine functions respectively, creating the characteristic oscillatory pattern.
Q3: Why does amplitude not affect the period of a simple harmonic oscillator?
The period and frequency of a simple harmonic oscillator are independent of amplitude. As amplitude changes, the maxima and minima of the waveform shift, but the oscillation rate remains constant. This property means a diving board's bounce frequency depends on its stiffness and mass, not how far it's pushed down.
Q4: What is the fundamental force relationship in simple harmonic motion?
In simple harmonic motion, acceleration and net force are proportional to displacement and act in the opposite direction. This restoring force characteristic drives the oscillatory behavior. The force constant determines how quickly the system responds, with stiffer systems having smaller periods than flexible ones.
Q5: How does mass affect the oscillation rate of a simple harmonic oscillator?
Mass directly influences the period of oscillation. A heavier person on a diving board bounces slower than a lighter person, demonstrating that increased mass increases the period. Combined with the force constant, mass determines the overall oscillation frequency of the system.
Q6: What happens to the waveform when you change the phase angle in simple harmonic motion?
Altering the phase angle shifts the cosine curve horizontally. Negative phase angle values shift the curve rightward, while positive values shift it leftward. This phase shift describes the time interval difference between two similar waveforms, measured in radians, without affecting the period or frequency.
Q7: How can you connect simple harmonic motion to circular motion?
Simple harmonic motion can be understood as the projection of uniform circular motion onto a diameter. The sinusoidal displacement, velocity, and acceleration patterns emerge naturally from this geometric relationship. Understanding simple harmonic motion and uniform circular motion together provides deeper insight into oscillatory systems.