Physics | Simple harmonic motion

C.1 Simple Harmonic Motion (SHM) - The Engine of Waves

Welcome to the study of Simple Harmonic Motion! This chapter is foundational. While it might seem like we are studying things that simply bounce, SHM is the critical link between Newton's mechanics and the world of waves (which is the major section we are in!). Understanding how a single particle oscillates allows us to understand how energy propagates through a medium—the definition of a wave.

Don't worry if the math looks complex initially. We will break down SHM into three simple ideas: definition, motion, and energy. Remember, if you can describe something swinging or vibrating, you are already halfway there!

What is Oscillation? (A Quick Review)

An oscillation or vibration is any repetitive movement around an equilibrium position. Think of a guitar string or a pendulum clock.

• Equilibrium Position: The point where the net force on the object is zero (the object would naturally rest here).
• Displacement (\(x\)): The distance of the oscillating object from the equilibrium position. It is a vector quantity, so direction matters.
• Amplitude (\(A\)): The maximum displacement from the equilibrium position. This defines the "size" of the oscillation.
• Period (\(T\)): The time taken for one complete oscillation (cycle). Measured in seconds (s).
• Frequency (\(f\)): The number of complete oscillations per unit time. Measured in Hertz (Hz), where \(1 \text{ Hz} = 1 \text{ s}^{-1}\).

Key Relationship: Period and frequency are inversely related:
\[T = \frac{1}{f}\]

1. Defining Simple Harmonic Motion (SHM)

Not all oscillations are SHM! SHM is a *specific* type of oscillation defined by a very precise relationship between acceleration and displacement.

The Defining Condition of SHM

Simple Harmonic Motion is defined as the motion of an oscillator whose acceleration is directly proportional to its displacement from the equilibrium position and is always directed towards the equilibrium position.

The Crucial Equation

This definition is captured mathematically by the defining equation of SHM:

\[a = -\omega^2 x\]

Where:

• \(a\) is the acceleration.
• \(x\) is the displacement from equilibrium.
• \(\omega\) (omega) is the angular frequency (a constant for a given system, measured in \(\text{rad s}^{-1}\)).

Why the Negative Sign?

The negative sign is the most important part! It means that the acceleration (\(a\)) is always in the opposite direction to the displacement (\(x\)).

Analogy: Imagine pulling a spring (positive \(x\)). The acceleration (and restoring force) points back towards the center (negative \(a\)). If the mass overshoots and is compressed (negative \(x\)), the acceleration points away from the wall (positive \(a\)).

This opposing acceleration creates a restoring force that always attempts to bring the object back to the center.

Relationship to Angular Frequency (\(\omega\))

Angular frequency (\(\omega\)) relates the time-based characteristics (\(T\) and \(f\)) to the oscillating system:

\[\omega = 2\pi f = \frac{2\pi}{T}\]

Since \(\omega\) is constant for a given SHM setup, the Period (\(T\)) and Frequency (\(f\)) are also constant. They do not depend on the amplitude \(A\).

Did you know? This principle is why grandfather clocks work! The clock's period remains the same even if the pendulum's swing gets slightly smaller as it runs down.

2. Kinematics of SHM: Displacement, Velocity, and Acceleration

Since SHM is a continuous, sinusoidal motion, the displacement, velocity, and acceleration are constantly changing, but they are all linked by the angular frequency, \(\omega\).

A. Displacement (\(x\))

We can describe the position of an object undergoing SHM using sinusoidal functions (sine or cosine). Assuming the motion starts at maximum displacement (\(x = A\) when \(t = 0\)):

\[x(t) = A \cos(\omega t)\]

If the motion started at the equilibrium position (\(x = 0\) when \(t = 0\)), we would use the sine function.

B. Velocity (\(v\))

Velocity is the rate of change of displacement. In SHM, the magnitude of velocity is maximum at the equilibrium position (\(x=0\)) and zero at the extremes (\(x=\pm A\)).

Maximum Velocity (\(v_{max}\)):

\[v_{max} = \omega A\]

Velocity at any displacement (\(x\)):

\[v = \pm \omega \sqrt{A^2 - x^2}\]

Note: You must choose the sign (\(\pm\)) based on whether the object is moving inward or outward.

C. Acceleration (\(a\))

Acceleration is defined by \(a = -\omega^2 x\). It is always directed toward the center.

Maximum Acceleration (\(a_{max}\)):

Acceleration is maximum when displacement is maximum (at the extremes, \(x=\pm A\)):

\[a_{max} = \omega^2 A\]

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Understanding Phase Differences Graphically

The key to understanding SHM graphs is noting the phase relationship (how much one quantity leads or lags another).

• When displacement (\(x\)) is max (extemes), velocity (\(v\)) is zero, and acceleration (\(a\)) is max (but opposite sign to \(x\)).
• When displacement (\(x\)) is zero (equilibrium), velocity (\(v\)) is max, and acceleration (\(a\)) is zero.

In terms of phase shift:

If \(x\) follows a cosine wave, \(v\) follows a negative sine wave (leads \(x\) by 90° or \(\pi/2\) radians), and \(a\) follows a negative cosine wave (leads \(x\) by 180° or \(\pi\) radians).

Memory Aid: If you are at the maximum displacement, you stop for a split second (\(v=0\)), but the force pulling you back is strongest (\(a=a_{max}\)).

3. Energy Changes in SHM

In an ideal SHM system (no air resistance or friction), the Total Energy of the system is conserved (remains constant).

The energy continuously transforms between Kinetic Energy (\(E_K\)) and Potential Energy (\(E_P\)).

Kinetic Energy (\(E_K\))

Kinetic energy is related to motion:

\[E_K = \frac{1}{2} m v^2\]

• \(E_K\) is maximum at the equilibrium position (\(x=0\)) where \(v\) is maximum.
• \(E_K\) is zero at the extremes (\(x=\pm A\)) where \(v\) is zero.

Potential Energy (\(E_P\))

Potential energy is the stored energy due to the object's position (due to the stretching or compressing of the spring, or the height of the pendulum mass).

• \(E_P\) is zero at the equilibrium position (\(x=0\)).
• \(E_P\) is maximum at the extremes (\(x=\pm A\)) where the system is maximally stretched or compressed.

Total Energy (\(E_T\))

The total mechanical energy is the sum of kinetic and potential energy, and it is constant:

\[E_T = E_K + E_P = \text{Constant}\]

Since the total energy must be equal to the maximum kinetic energy (when \(E_P = 0\)) or the maximum potential energy (when \(E_K = 0\)), we can define the total energy using the amplitude \(A\).

For any system where the restoring force is proportional to displacement (like a mass on a spring, where \(F = kx\)), the maximum potential energy stored is \(E_{P, max} = \frac{1}{2} k A^2\). Therefore:

\[E_T = \frac{1}{2} k A^2\]

Key Takeaway: The Total Energy of an SHM system is proportional to the square of the amplitude (\(E_T \propto A^2\)). Double the amplitude, and you quadruple the energy!

4. Examples of SHM Systems

While the principles of SHM are universal, the expressions for period (\(T\)) depend on the physical constants of the specific system.

A. Mass on a Spring System (Horizontal or Vertical)

For a mass \(m\) attached to a spring with spring constant \(k\), the restoring force is given by Hooke's Law: \(F = -kx\).

Using Newton's Second Law (\(F=ma\)) and the defining equation (\(a = -\omega^2 x\)), we find that:

\[\omega^2 = \frac{k}{m}\]

Thus, the Period (\(T\)) is:

\[T = 2\pi \sqrt{\frac{m}{k}}\]

• Important: The period only depends on the mass (\(m\)) and the stiffness of the spring (\(k\)). It does NOT depend on the amplitude (\(A\)) or gravity (\(g\)).

B. Simple Pendulum (Small Angle Approximation)

A simple pendulum (a point mass \(m\) suspended by a string of length \(L\)) only undergoes true SHM if the angle of swing is very small (typically less than 10°).

The Period (\(T\)) for a simple pendulum is:

\[T = 2\pi \sqrt{\frac{L}{g}}\]

• Important: The period only depends on the length (\(L\)) of the string and the acceleration due to gravity (\(g\)). It does NOT depend on the mass (\(m\)) or the amplitude (\(A\)), provided the angle is small.
• Common Mistake to Avoid: Remember that a large-angle pendulum swing is not SHM. This is because the restoring force is proportional to \(\sin\theta\), not just \(\theta\) (or displacement \(x\)).