C.1.1 Defining and analyzing simple harmonic motion

Conditions for Simple Harmonic Motion (SHM)

Simple Harmonic Motion (SHM) is a special type of oscillatory motion where an object moves back and forth around an equilibrium position.

Restoring Force Proportional to Displacement

For SHM to occur, the system must have a restoring force that:

Acts in the opposite direction to the displacement.
Is directly proportional to the displacement from the equilibrium position.

This relationship can be expressed mathematically as:

$F = - k x$

where:

$F$ is the restoring force.
$k$ is a constant (often called the spring constant in a spring-mass system).
$x$ is the displacement from the equilibrium position.

Hint

The negative sign indicates that the force acts in the opposite direction to the displacement.

Example

In a mass-spring system, when the mass is displaced to the right, the spring exerts a force to the left, trying to bring the mass back to equilibrium.
This force is proportional to how far the mass is stretched or compressed.

Acceleration in SHM

The restoring force causes the object to accelerate back towards the equilibrium position.

Using Newton’s second law, $F = m a$ , we can express the acceleration as:

$a = - \frac{k}{m} x$

This shows that the acceleration is also proportional to the displacement and acts in the opposite direction.

Graph showing proportionality between acceleration an displacement.

Defining Equation for SHM: $a = - ω^{2} x$

The defining equation for SHM is:

$a = - ω^{2} x$

where:

$a$ is the acceleration.
$x$ is the displacement.
$ω$ (omega) is the angular frequency, a constant that characterizes the system.

Hint

The negative sign indicates that the acceleration is always directed opposite to the displacement, ensuring the object is pulled back towards equilibrium.

Why $a = - ω^{2} x$ ?

The term $ω^{2}$ is derived from the system’s properties.
In a mass-spring system, for example, $ω^{2} = \frac{k}{m}$ , where $k$ is the spring constant and $m$ is the mass.

This relationship highlights how the system’s physical characteristics determine its oscillatory behavior.

Tip

The negative sign in $a = - ω^{2} x$ is crucial. It ensures that the acceleration always opposes the displacement, a fundamental requirement for SHM.

Key Parameters of SHM

To fully describe SHM, we need to understand several key parameters:

1. Amplitude, Equilibrium Position, and Displacement

Amplitude ( $A$ ):
- The maximum displacement from the equilibrium position.
- It represents the furthest point the object reaches during its oscillation.
Equilibrium Position:
- The point where the net force on the object is zero.
- In SHM, the object oscillates symmetrically around this position.
Displacement ( $x$ ):
- The distance of the object from the equilibrium position at any given time.
- Displacement can be positive or negative, depending on the direction.

Example

If a pendulum swings 5 cm to the right and 5 cm to the left of its equilibrium position, its amplitude is 5 cm.
The displacement varies between +5 cm and -5 cm during the oscillation.

2. Time Period ( $T$ ), Frequency ( $f$ ), and Angular Frequency ( $ω$ )

These parameters describe the timing of the oscillation:

Time Period ( $T$ ): The time taken to complete one full oscillation. It is measured in seconds (s).
Frequency ( $f$ ): The number of oscillations per second. It is measured in hertz (Hz) and is the inverse of the period:

$f = \frac{1}{T}$

Angular Frequency ( $ω$ ): A measure of how quickly the object oscillates, expressed in radians per second (rad/s). It is related to the period and frequency by:

$ω = 2 π f = \frac{2 π}{T}$

Example

A mass-spring system with a period of 2 seconds has a frequency of 0.5 Hz and an angular frequency of: $ω = 2 π \times 0.5 = π rad/s$

Tip

Angular frequency ( $ω$ ) is especially useful in SHM because it connects the motion to circular motion, where $ω$ describes how fast an object moves around a circle.

Why SHM is Fundamental

SHM is not just a theoretical concept; it underpins many real-world phenomena.

Example

The oscillations of a guitar string, the vibrations of atoms in a solid, and the motion of a pendulum clock all exhibit SHM or can be approximated as SHM under certain conditions.

Reflection

Theory of Knowledge

How does the simplicity of SHM help physicists model more complex oscillatory systems?
Can you think of other areas in science where a simple model provides insights into more complicated phenomena?

Self review

What are the conditions for a system to exhibit SHM?
How is the angular frequency ( $ω$ ) related to the period ( $T$ ) and frequency ( $f$ )?
Why is the acceleration in SHM always directed opposite to the displacement?

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C.1.1 Defining and analyzing simple harmonic motion

Conditions for Simple Harmonic Motion (SHM)

Restoring Force Proportional to Displacement

Acceleration in SHM

Defining Equation for SHM: a=−ω2x

Why a=−ω2x?

Key Parameters of SHM

1. Amplitude, Equilibrium Position, and Displacement

2. Time Period (T), Frequency (f), and Angular Frequency (ω)

Why SHM is Fundamental

Reflection

You've read 2/2 free chapters this week.

Simple Harmonic Motion (SHM)

Defining Equation for SHM: $a = - ω^{2} x$

Why $a = - ω^{2} x$ ?

2. Time Period ( $T$ ), Frequency ( $f$ ), and Angular Frequency ( $ω$ )