Periodic Motion — Revision Notes

⚡ 30-Second Revision

Periodic Motion — Repeats after fixed time $T$.
Period ($T$) — Time for one cycle (s).
Frequency ($f$) — Cycles per second ($f = 1/T$) (Hz).
Angular Frequency ($\omega$) — $\omega = 2\pi f = 2pi/T$ (rad/s).
Oscillatory Motion — Periodic + to-and-fro about mean position.
Simple Harmonic Motion (SHM) — Oscillatory + restoring force $F \propto -x$.
Pendulum Period (small $\theta$) — $T = 2\pi \sqrt{\frac{L}{g}}$.
Spring-Mass Period — $T = 2\pi \sqrt{\frac{m}{k}}$.
Hierarchy — SHM $\subset$ Oscillatory $\subset$ Periodic.

2-Minute Revision

Periodic motion is any motion that repeats itself identically after a fixed time interval, called the period ($T$). The frequency ($f$) is the number of repetitions per unit time, related by $f = 1/T$.

Angular frequency ($\omega$) is $2\pi f$ or $2pi/T$. Key examples include uniform circular motion (periodic but not oscillatory), a simple pendulum (periodic and oscillatory), and a mass on a spring (periodic, oscillatory, and simple harmonic).

It's crucial to understand the hierarchy: all SHM is oscillatory, and all oscillatory motion is periodic, but the reverse is not true. For a simple pendulum (small angles), $T = 2\pi \sqrt{L/g}$, meaning $T \propto \sqrt{L}$.

For a spring-mass system, $T = 2\pi \sqrt{m/k}$, meaning $T \propto \sqrt{m}$ and $T \propto 1/\sqrt{k}$. Remember to convert units (e.g., minutes to seconds) when calculating frequency or period.

5-Minute Revision

Periodic motion is the most general type of repetitive motion, characterized by its repetition after a fixed time interval, the period ($T$). Its reciprocal is frequency ($f = 1/T$), which measures cycles per second (Hz). Angular frequency ($\omega = 2\pi f$) is also used, especially for oscillations, and is measured in radians per second. A key distinction for NEET is the hierarchy of repetitive motions:

1
Periodic Motion — Broadest. Examples: Earth's orbit, fan rotation, pendulum swing.
2
Oscillatory Motion — A subset of periodic motion where the object moves back and forth about a mean position. Examples: Simple pendulum, vibrating string.
3
Simple Harmonic Motion (SHM) — A subset of oscillatory motion where the restoring force is directly proportional to displacement from equilibrium ($F \propto -x$). Examples: Ideal spring-mass system, simple pendulum at small angles.

Key Formulas to Recall:

For a simple pendulum (small angles): $T = 2\pi \sqrt{\frac{L}{g}}$. This implies $T$ is independent of mass and amplitude (for small angles), and $T \propto \sqrt{L}$.
For a spring-mass system: $T = 2\pi \sqrt{\frac{m}{k}}$. This implies $T$ is independent of amplitude, and $T \propto \sqrt{m}$, $T \propto 1/\sqrt{k}$.

Worked Example: A spring-mass system has a period of $1.0\,\text{s}$. If the mass is doubled and the spring constant is quadrupled, what is the new period? Solution: Original period $T_1 = 2\pi \sqrt{\frac{m}{k}} = 1.

0\,\text{s}$. New mass$m' = 2m$. New spring constant$k' = 4k$. New period$T_2 = 2\pi \sqrt{\frac{m'}{k'}} = 2\pi \sqrt{\frac{2m}{4k}} = 2\pi \sqrt{\frac{1}{2} \frac{m}{k}} = \frac{1}{\sqrt{2}} \left(2\pi \sqrt{\frac{m}{k}}\right) = \frac{T_1}{\sqrt{2}}$.

So, $T_2 = \frac{1.0\,\text{s}}{\sqrt{2}} \approx 0.707\,\text{s}$.

Always be careful with units and the square root relationships. Conceptual questions often test your ability to differentiate between these motion types and their specific conditions.

Prelims Revision Notes

Periodic motion is defined as any motion that repeats itself identically after a fixed interval of time, called the **period ($T$). Its SI unit is seconds (s). The frequency ($f$)** is the number of cycles per unit time, and it is the reciprocal of the period: $f = 1/T$.

The SI unit for frequency is Hertz (Hz), where $1\,\text{Hz} = 1\,\text{s}^{-1}$. **Angular frequency ($\omega$)** is related to frequency by $\omega = 2\pi f = 2pi/T$, with units of radians per second (rad/s).

These three quantities are fundamental to describing any repetitive motion.

It's crucial to distinguish between different types of repetitive motion:

Periodic Motion — Broadest category. Examples: Earth's orbit, rotation of a fan, hands of a clock. Uniform circular motion is periodic but NOT oscillatory.
Oscillatory Motion — A subset of periodic motion where the object moves 'to and fro' about a fixed equilibrium position. All oscillatory motions are periodic. Examples: Simple pendulum (any amplitude), vibrating string.
Simple Harmonic Motion (SHM) — A special case of oscillatory motion where the restoring force is directly proportional to the displacement from equilibrium and always directed towards it ($F = -kx$). All SHMs are oscillatory and thus periodic. Examples: Mass on an ideal spring, simple pendulum for small angles.

Key Formulas for Period:

Simple Pendulum (small angles) — $T = 2\pi \sqrt{\frac{L}{g}}$

* $T$ is independent of mass and amplitude (for small angles). * $T \propto \sqrt{L}$ (period increases with length). * $T \propto 1/\sqrt{g}$ (period decreases with increasing gravity).

Spring-Mass System — $T = 2\pi \sqrt{\frac{m}{k}}$

* $T$ is independent of amplitude. * $T \propto \sqrt{m}$ (period increases with mass). * $T \propto 1/\sqrt{k}$ (period decreases with stiffer spring).

Remember to convert time units (e.g., minutes to seconds) when calculating frequency or period. Pay attention to conceptual questions that test the hierarchy and specific conditions for each type of motion. For instance, a common trap is assuming all periodic motion is SHM.

Vyyuha Quick Recall

To remember the hierarchy of repetitive motions: People Often Sing Hymns. Periodic (broadest) -> Oscillatory (back & forth) -> Simple Harmonic (force $\propto$ displacement).