Simple harmonic motion

Simple harmonic motion (SHM) is a periodic oscillatory motion where the restoring force is directly proportional to the displacement from the equilibrium position and acts in the direction opposite to the displacement.

Mathematically, Hooke's Law describes this restoring force:
$F = -k x$
where:

• $F$ is the restoring force in newtons ($\text{N}$)

• $k$ is the stiffness or spring constant in newtons per meter ($\text{N/m}$)

• $x$ is the displacement from equilibrium in meters ($\text{m}$)

Differential Equation of Motion

Applying Newton's second law of motion:
$F = m a$
$m rac{d^2 x}{dt^2} = -k x$
$rac{d^2 x}{dt^2} + rac{k}{m} x = 0$

Defining the angular frequency $\omega$ as:
$\omega = \sqrt{\frac{k}{m}}$
$\frac{d^2 x}{dt^2} + \omega^2 x = 0$

Solving the Differential Equation

This is a second-order linear homogeneous differential equation. The general solution for displacement $x(t)$ is:
$x(t) = A \cos(\omega t + \phi)$
where:

• $A$ is the amplitude of oscillation ($\text{m}$)

• $\omega$ is the angular frequency ($\text{rad/s}$)

• $t$ is time ($\text{s}$)

• $\phi$ is the phase constant or phase angle ($\text{rad}$)

Velocity and Acceleration Expressions

1. Displacement:
   $x(t) = A \cos(\omega t + \phi)$

2. Velocity:
   $v(t) = \frac{dx}{dt} = -A \omega \sin(\omega t + \phi)$
   The maximum velocity magnitude is:
   $v_{\max} = A \omega$

3. Acceleration:
   $a(t) = \frac{d^2 x}{dt^2} = -A \omega^2 \cos(\omega t + \phi) = -\omega^2 x(t)$
   The maximum acceleration magnitude is:
   $a_{\max} = A \omega^2$

Period and Frequency

• Period ($T$): The time taken to complete one full oscillation cycle.
  $T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{m}{k}}$

• Frequency ($f$): The number of complete cycles per unit time in hertz ($\text{Hz}$).
  $f = \frac{1}{T} = \frac{\omega}{2\pi} = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$

Energy in Simple Harmonic Motion

In an ideal conservative system, total mechanical energy is conserved and continuously converts between kinetic energy ($E<em>k$) and potential energy ($E</em>p$).

1. Potential Energy:
   $E_p = \frac{1}{2} k x^2 = \frac{1}{2} k A^2 \cos^2(\omega t + \phi)$

2. Kinetic Energy:
   $E_k = \frac{1}{2} m v^2 = \frac{1}{2} m A^2 \omega^2 \sin^2(\omega t + \phi) = \frac{1}{2} k A^2 \sin^2(\omega t + \phi)$

3. Total Energy ($E<em>{\text{total}}$):
   $E</em>{\text{total}} = E<em>p + E</em>k = \frac{1}{2} k A^2 \left(\cos^2(\omega t + \phi) + \sin^2(\omega t + \phi)\right) = \frac{1}{2} k A^2$

The total mechanical energy is constant and directly proportional to the square of the amplitude $A$.

Simple harmonic motion (SHM) is a periodic oscillatory motion where the restoring force is directly proportional to the displacement from the equilibrium position and acts in the direction opposite to the displacement.

1. Fundamental Definition and Hooke's Law

Mathematically, Hooke's Law describes this restoring force:
$F = -k x$
where:

• $F$ is the restoring force in newtons ($\text{N}$)

• $k$ is the stiffness or spring constant in newtons per meter ($\text{N/m}$)

• $x$ is the displacement from equilibrium in meters ($\text{m}$)

2. Differential Equation of Motion

Applying Newton's second law of motion ($F = m a$):
$m \frac{d^2 x}{dt^2} = -k x$
$\frac{d^2 x}{dt^2} + \frac{k}{m} x = 0$

Defining the angular frequency $\omega$ as:
$\omega = \sqrt{\frac{k}{m}}$
$\frac{d^2 x}{dt^2} + \omega^2 x = 0$

3. Solving the Differential Equation

This is a second-order linear homogeneous differential equation with constant coefficients.

Assuming a trial solution of the form $x(t) = C e^{r t}$:
$r^2 + \omega^2 = 0 \implies r = \pm i \omega$

Using Euler's formula, the general real-valued solution for displacement $x(t)$ is:
$x(t) = A \cos(\omega t + \phi)$
where:

• $A$ is the amplitude of oscillation ($\text{m}$)

• $\omega$ is the angular frequency ($\text{rad/s}$)

• $t$ is time ($\text{s}$)

• $\phi$ is the phase constant or phase angle ($\text{rad}$)

4. Kinematics of Simple Harmonic Motion

1. Displacement:
   $x(t) = A \cos(\omega t + \phi)$

2. Velocity:
   $v(t) = \frac{dx}{dt} = -A \omega \sin(\omega t + \phi)$
   The maximum velocity magnitude occurs at $x = 0$:
   $v_{\max} = A \omega$
   Velocity expressed in terms of displacement $x$:
   $v(x) = \pm \omega \sqrt{A^2 - x^2}$

3. Acceleration:
   $a(t) = \frac{d^2 x}{dt^2} = -A \omega^2 \cos(\omega t + \phi) = -\omega^2 x(t)$
   The maximum acceleration magnitude occurs at $x = \pm A$:
   $a_{\max} = A \omega^2$

5. Mass-Spring Systems

5.1 Horizontal Spring System

• Equilibrium position occurs where the spring is uncompressed ($x = 0$).

• Angular frequency:
  $\omega = \sqrt{\frac{k}{m}}$

5.2 Vertical Spring System

• At rest, gravity is balanced by the spring force:
  $m g = k \Delta L$

• Displacement relative to the new equilibrium position $y_0$ yields the same equation of motion:
  $\frac{d^2 y}{dt^2} + \omega^2 y = 0$

6. Simple Pendulum Derivation

A simple pendulum consists of a point mass $m$ suspended from a massless string of length $L$.

Applying Newton's second law for tangential force:
$F_{\text{tangential}} = -m g \sin(\theta)$
$m L \frac{d^2 \theta}{dt^2} = -m g \sin(\theta)$
$\frac{d^2 \theta}{dt^2} + \frac{g}{L} \sin(\theta) = 0$

Small-Angle Approximation

For small angles where $\theta \ll 1\,\text{rad}$, $\sin(\theta) \approx \theta$:
$\frac{d^2 \theta}{dt^2} + \frac{g}{L} \theta = 0$

The angular frequency and period are:
$\omega = \sqrt{\frac{g}{L}}$
$T = 2\pi \sqrt{\frac{L}{g}}$

7. Period and Frequency

• Period ($T$): The time taken to complete one full oscillation cycle.
  $T = \frac{2\pi}{\omega} = 2\pi \sqrt{\frac{m}{k}}$

• Frequency ($f$): The number of complete cycles per unit time in hertz ($\text{Hz}$).
  $f = \frac{1}{T} = \frac{\omega}{2\pi} = \frac{1}{2\pi} \sqrt{\frac{k}{m}}$

8. Energy in Simple Harmonic Motion

In an ideal conservative system, total mechanical energy is conserved and continuously converts between kinetic energy ($E<em>k$) and potential energy ($E</em>p$).

1. Potential Energy:
   $E_p = \frac{1}{2} k x^2 = \frac{1}{2} k A^2 \cos^2(\omega t + \phi)$

2. Kinetic Energy:
   $E_k = \frac{1}{2} m v^2 = \frac{1}{2} m A^2 \omega^2 \sin^2(\omega t + \phi) = \frac{1}{2} k A^2 \sin^2(\omega t + \phi)$

3. Total Energy ($E<em>{\text{total}}$):
   $E</em>{\text{total}} = E<em>p + E</em>k = \frac{1}{2} k A^2 \left(\cos^2(\omega t + \phi) + \sin^2(\omega t + \phi)\right) = \frac{1}{2} k A^2$

9. Damped Oscillations

When resistive forces like friction or viscous drag ($F<em>{\text{drag}} = -b v$) act on the system, the equation of motion becomes: $m \frac{d^2 x}{dt^2} + b \frac{dx}{dt} + k x = 0$ $\frac{d^2 x}{dt^2} + 2\gamma \frac{dx}{dt} + \omega</em>0^2 x = 0$
where $\gamma = \frac{b}{2m}$ is the damping coefficient and $\omega_0 = \sqrt{\frac{k}{m}}$ is the natural angular frequency.

Depending on $\gamma$ relative to $\omega_0$:

• Underdamped ($\gamma < \omega_0$): System oscillates with exponentially decreasing amplitude.

• Critically damped ($\gamma = \omega_0$): System returns to equilibrium as quickly as possible without oscillating.

• Overdamped ($\gamma > \omega_0$): System returns to equilibrium slowly without oscillating.

10. Forced Oscillations and Resonance

When a periodic external driving force $F(t) = F<em>0 \cos(\omega</em>d t)$ is applied:
$m \frac{d^2 x}{dt^2} + b \frac{dx}{dt} + k x = F<em>0 \cos(\omega</em>d t)$

At steady state, the amplitude of oscillation depends on the driving frequency $\omega<em>d$: $A(\omega</em>d) = \frac{F<em>0}{\sqrt{m^2 (\omega</em>0^2 - \omega<em>d^2)^2 + b^2 \omega</em>d^2}}$

Resonance occurs when the driving frequency matches the natural frequency ($\omega<em>d \approx \omega</em>0$), resulting in a maximum amplitude of oscillation.