A simple pendulum is an idealized model consisting of a point mass (called the bob) suspended from a fixed pivot by a massless, inextensible string or rod. When displaced from its equilibrium position (hanging straight down) and released, it swings back and forth due to gravity.

Key Concepts:

1. Restoring Force:
   • When the pendulum bob is displaced, gravity pulls it downwards. However, the tension in the string pulls it towards the pivot. The combination of these two forces results in a net force that always acts to bring the bob back to its equilibrium position. This is called the restoring force.
   • For small angles of displacement (typically less than about 10-15 degrees), the restoring force is approximately proportional to the displacement. This is crucial for Simple Harmonic Motion (SHM).
2. Simple Harmonic Motion (SHM):
   • A simple pendulum approximates SHM when its initial angle of displacement is small. In SHM, the restoring force is directly proportional to the displacement and acts in the opposite direction.
   • Systems undergoing SHM exhibit a periodic, oscillating motion, returning to their original position and velocity after a fixed amount of time (the period).
3. Period (T):
   • The period of a pendulum is the time it takes for one complete back-and-forth swing (one oscillation).
   • For a simple pendulum undergoing SHM (small angles), a remarkable property is that its period does not depend on the mass of the bob or the initial amplitude (angle) of the swing. It only depends on its length and the local gravitational acceleration.
4. Frequency (f):
   • Frequency is the number of complete swings (oscillations) per unit time. It's the inverse of the period: $f = 1/T$.
5. Damping (Air Friction):
   • In a real-world scenario (and in your simulation), the pendulum eventually stops swinging. This is due to damping forces, such as air resistance (friction) and friction at the pivot.
   • Damping is a force that opposes the motion and is often proportional to the velocity of the object. It causes the amplitude of the oscillations to gradually decrease over time.

Relevant Formulas:

For a Simple Pendulum (assuming small angles and no damping):

1. Period (T):

   The period of a simple pendulum is given by:

   $$T = 2\pi \sqrt{\frac{L}{g}}$$

   Where:

   • $T$ is the period (in seconds, s)
   • $\pi$ (pi) is a mathematical constant, approximately 3.14159
   • $L$ is the length of the pendulum (from the pivot to the center of the bob, in meters, m)
   • $g$ is the acceleration due to gravity (in meters per second squared, m/s²)

   What this tells us:

   • A longer pendulum has a longer period (swings slower).
   • Stronger gravity makes the pendulum swing faster (shorter period).
2. Angular Frequency ($\omega$):

   The angular frequency is related to the period and frequency:

   $$\omega = \frac{2\pi}{T} = 2\pi f$$

   From the period formula, for a simple pendulum:

   $$\omega = \sqrt{\frac{g}{L}}$$

   Where $\omega$ is in radians per second (rad/s).

3. Equation of Motion (for small angles):

   The angular displacement $\theta$ (in radians) of a simple pendulum as a function of time $t$ can be described by:

   $$\theta(t) = \theta_0 \cos(\omega t + \phi)$$

   Where:

   • $\theta(t)$ is the angular displacement at time $t$
   • $\theta_0$ is the initial maximum angular displacement (amplitude)
   • $\omega$ is the angular frequency
   • $\phi$ is the phase constant (depends on the initial conditions, often 0 if released from rest at $\theta_0$)
4. Tangential velocity ($v$) The tangential velocity $v$ is related to the angular velocity $\dot{\theta}$ by $v = L\dot{\theta}$. Differentiating the angular displacement: $$\dot{\theta}(t) = -\omega \theta_{max} \sin(\omega t + \phi)$$So, the velocity of the bob is:$$v(t) = L \dot{\theta}(t) = -L\omega \theta_{max} \sin(\omega t + \phi)$$Substituting $\omega = \sqrt{\frac{g}{L}}$:$$v(t) = -L\sqrt{\frac{g}{L}} \theta_{max} \sin(\omega t + \phi)$$ $$v(t) = -\sqrt{gL} \theta_{max} \sin(\omega t + \phi)$$ The maximum velocity in SHM occurs when $\sin(\omega t + \phi) = \pm 1$: $$v_{max} = \sqrt{gL} \theta_{max}$$ (where $\theta_{max}$ is in radians).
5. What happens with Damping:

   When damping is introduced (like the "Air Friction" slider in your simulation), the simple SHM formulas are no longer perfectly accurate because energy is being dissipated from the system.

   • The amplitude of the swing will decrease over time.
   • The motion is no longer purely periodic in the strict sense, as the energy is dying out.
   • The equation of motion becomes more complex, often involving an exponential decay term:

     $$\theta(t) = \theta_0 e^{-\gamma t} \cos(\omega' t + \phi)$$

     Where:

     • $e$ is Euler's number (approx. 2.718)
     • $\gamma$ (gamma) is the damping coefficient (related to your "Air Friction" setting)
     • $\omega'$ is the damped angular frequency, which is slightly less than the undamped $\omega$.

Reflection