15.1
Simple Harmonic Motion
Simple harmonic motion is the name given to oscillatory motion for a system where the net force can be described by Hooke's law. If the net force can…
If moving objects repeat their path over a regular interval, it is called harmonic or periodic motion. Periodic motion in which an object moves to and fro from a fixed point is called oscillation.
If the restoring force on the object is proportional to the displacement from the equilibrium position, the oscillation is called simple harmonic motion. However, not all periodic oscillations are simple harmonic motions.
Consider a ruler with one end fixed to the tabletop and the other free with a mass attached. When this system is pulled up and released, it vibrates.
Here, the acceleration is directed toward the center of oscillation and is proportional to the displacement from the equilibrium position. As a result, the force is proportional to the displacement but in the opposite direction. The proportionality constant is the spring constant.
One complete oscillation is called a cycle, and the time required to complete one cycle is called a period. The number of cycles per unit time is the frequency.
At equilibrium, the net force is zero. The maximum displacement from the equilibrium position is the amplitude.
View the full transcript and gain access to JoVE Core videos
Q1: What distinguishes simple harmonic motion from other types of periodic motion?
Simple harmonic motion occurs when the restoring force on an object is proportional to its displacement from equilibrium, following Hooke's law. Not all periodic oscillations meet this criterion. In simple harmonic motion, acceleration is directed toward the center of oscillation and is proportional to displacement. This proportional relationship, governed by the spring constant, defines the motion's character and makes it predictable and mathematically elegant.
Q2: How are period and frequency related in simple harmonic motion?
Period is the time required to complete one cycle, while frequency is the number of cycles per unit time. These quantities have an inverse relationship: frequency equals one divided by period. Neither period nor frequency depends on amplitude. The SI unit for frequency is Hertz. This inverse relationship allows you to convert between the two measurements using simple mathematical operations.
Q3: What is amplitude and how does it affect oscillation characteristics?
Amplitude is the maximum displacement from the equilibrium position. In simple harmonic motion, amplitude does not affect the period or frequency of oscillation. This independence means that whether an object oscillates with small or large displacement, it completes each cycle in the same time. This property is fundamental to understanding why simple harmonic oscillators maintain consistent timing regardless of how far they move.
Q4: What happens to the net force at equilibrium in simple harmonic motion?
At equilibrium, the net force on the oscillating object is zero. This is the central reference point around which the object oscillates. When displaced from equilibrium, a restoring force proportional to the displacement acts to return the object to this position. The equilibrium position represents the balance point where all forces cancel, making it the natural resting state for the system.
Q5: How do you calculate frequency if you know the period of oscillation?
Frequency is calculated by taking the inverse of the period. If a medical imaging device produces ultrasound with a period of 0.400 microseconds, the frequency equals one divided by 0.400 microseconds, yielding 2.5 megahertz. This inverse relationship applies to all oscillatory systems. Conversely, if you know frequency, divide one by that value to find the period.
Q6: What role does the spring constant play in simple harmonic motion?
The spring constant is the proportionality constant that relates the restoring force to displacement from equilibrium. A stiffer spring has a larger constant and produces a stronger restoring force for the same displacement. The spring constant determines how quickly the system oscillates and influences the period and frequency of motion. Different spring constants result in different oscillation rates for the same mass.
Q7: Why is simple harmonic motion important in understanding oscillatory systems?
Simple harmonic motion provides a mathematical framework for analyzing many real-world oscillatory systems, from musical instruments to medical imaging devices. Understanding the relationship between force, displacement, and acceleration in simple harmonic motion enables prediction of system behavior. The principles apply broadly to springs, pendulums, and waves, making it foundational for advanced physics and engineering applications.