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In this post, we will be exploring three cases of simple harmonic motion. We will look at a horizontal, spring-block oscillator; a vertical, spring-block oscillator; and a simple pendulum.
Table of Contents
What is Simple Harmonic Motion?
Simple harmonic motion (SHM) is a special type of back and forth motion where the object oscillates around an equilibrium position. Key characteristics of SHM include a restoring force proportional to displacement, sinusoidal motion, and constant change in energy. The restoring force is a force that brings an object back into an equilibrium position, and the force is proportional to the displacement from this equilibrium position. Graphing the object’s position, velocity, and acceleration will all result in a sin or cos shaped graph. The energy of the object constantly changed from kinetic to potential and vice versa.
Demo 1 (Horizontal Spring-Block Oscillator)
Restoring Force (Fr)
The restoring Force is equal to the force of Spring (Fs).
$$Fr=Fs=-kx$$
- K: spring constant (N/m)
- x: displacement from equilibrium (m)
- The negative sign indicates the force always points toward equilibrium
Amplitude (A)
Amplitude is the maximum displacement from equilibrium.
Period (T)
Period is the time it takes to complete a cycle.
$$T=\frac{1}{f}$$
$$T=2π\sqrt\frac{m}{k}$$
Frequency (f)
Frequency is the number of cycles that can be completed per second.
$$f=\frac{1}{T}$$
$$f=\frac{1}{2π}\sqrt\frac{k}{m}$$
Energy
Total Mechanical Energy = Potential Energy of Spring + Kinetic Energy
$$E=SPE+KE=\frac{1}{2}kx^2+\frac{1}{2}mv^2$$
Summary
| Equilibrium Position | Endpoint Position | |
| Magnitude of Restoring Force | 0 | max |
| Magnitude of Acceleration | 0 | max |
| Potential Energy of Spring | 0 | max |
| Kinetic Energy | max | 0 |
| Speed of Block | max | 0 |
Demo 2 (Vertical Spring-Block Oscillator)
Mass on an unstretched spring dropped from rest
Restoring Force (Fr)
$$Fr=Fs-mg$$
Energy
Total Mechanical Energy = Gravitational Potential Energy + Potential Energy of Spring+ Kinetic Energy
$$E=GPE+SPE+KE=mgh+\frac{1}{2}kx^2+\frac{1}{2}mv^2$$
Summary
| Top Position | Equilibrium Position | Bottom Position | |
| Magnitude of Force of Spring | 0 | mg | max |
| Magnitude of Restoring Force | max | 0 | max |
| Magnitude of Acceleration | max | 0 | max |
| Gravitational Potential Energy | max | 0 | |
| Potential Energy of Spring | 0 | max | |
| Kinetic Energy | 0 | max | 0 |
| Speed of Ball | 0 | max | 0 |
Demo 3 (Simple Pendulum)
Restoring Force (Fr)
$$Fr=mgsinθ$$
Central Force (Fc)
$$Fc=\frac{mv^2}{L}$$
Tension (T)
Since
$$Fc=T-mgcosθ$$
We have
$$T=mgcosθ+\frac{mv^2}{L}$$
Period (T)
If θ is small, then sinθ ≈ θ. The magnitude of the restoring force is approximately mgθ. So the motion can be treated as simple harmonic. The period of the oscillations is:
$$T=2π\sqrt\frac{L}{g}$$
Frequency (f)
$$f=\frac{1}{2π}\sqrt\frac{g}{L}$$
Energy
Total Mechanical Energy = Potential Energy + Kinetic Energy
$$E=PE+KE=mgh+\frac{1}{2}mv^2$$
Summary
| Equilibrium Position | Endpoint Position | |
| Magnitude of Tension | max | min |
| Magnitude of Central Force | max | 0 |
| Magnitude of Restoring Force | 0 | max |
| Magnitude of Tangential Acceleration | 0 | max |
| Potential Energy | 0 | max |
| Kinetic Energy | max | 0 |
| Speed of Ball | max | 0 |
