### Notes from Toppers

**Detailed Notes from Toppers: Introduction to Periodic Motion**

**Simple Harmonic Motion (SHM)** *(NCERT Book: Physics Part 1, Class 11)*

**Definition:**SHM is a special case of periodic motion where the restoring force is directly proportional to the negative displacement from the mean position.**Mathematical Representation:**- Displacement: $$x = A\cos(\omega t + \phi)$$
- Velocity: $$v = -\omega A\sin(\omega t + \phi)$$
- Acceleration: $$a = -\omega^2 A\cos(\omega t + \phi)$$
- Where A is the amplitude, ω is the angular frequency, t is the time, and φ is the phase angle.

**Characteristics:**- The motion is periodic with a definite time period (T) and frequency (f).
- The acceleration of the particle is always directed toward the mean position.
- The velocity and acceleration are π/2 phase out of phase with each other.

**Equations of Motion in SHM** *(NCERT Book: Physics Part 1, Class 11)*

**Derivation of Equations:**- Starting with the definition of SHM, use Newton’s second law and the concept of angular frequency to derive the equations of motion.

**Applications:**- The equations of motion are used to determine the displacement, velocity, and acceleration of a particle undergoing SHM at any instant of time.
- They are also used to study the phase relationships between displacement, velocity, and acceleration.

**Energy in SHM** *(NCERT Book: Physics Part 1, Class 11)*

**Potential Energy:**The potential energy (U) of a particle in SHM is given by $$U = \frac{1}{2}kA^2\cos^2(\omega t + \phi)$$ where k is the spring constant.**Kinetic Energy:**The kinetic energy (K) of a particle in SHM is given by $$K = \frac{1}{2}kA^2\sin^2(\omega t + \phi)$$**Total Energy:**The total energy (E) of a particle in SHM is constant and is given by $$E = K + U = \frac{1}{2}kA^2$$

**Applications of SHM** *(NCERT Book: Physics Part 1, Class 11)*

**Real-World Examples:**SHM is found in various systems such as pendulums, springs, oscillating masses, sound waves, and alternating current circuits.**Resonance:**Resonance occurs when the frequency of an external force matches the natural frequency of a system, resulting in a significant increase in amplitude.**Damped Harmonic Motion:**When a resistive force opposes the motion, it is known as damped harmonic motion.**Forced Harmonic Motion:**When an external force is applied to a system, causing it to oscillate, it is called forced harmonic motion.

**Measurement of Time** *(NCERT Book: Physics Part 2, Class 12)*

**Pendulums and Quartz Clocks:**SHM is used in devices like pendulums and quartz clocks to measure time accurately.**Working Principle:**In a pendulum, the periodic motion of the bob is utilized, while in a quartz clock, the vibrations of a quartz crystal are used to regulate the timekeeping.

**Graphs and Problem-Solving** *(NCERT Book: Physics Part 1, Class 11)*

**Interpretation of Graphs:**Displacement-time, velocity-time, and acceleration-time graphs provide insights into the motion and energy variations in SHM.**Phase Relationships:**Graphs help analyze the phase relationships between displacement, velocity, and acceleration.**Problem-Solving:**Mathematical concepts and formulas are applied to solve numerical problems involving SHM, such as finding the amplitude, time period, frequency, and energy of the system.

Following these detailed notes, studying the relevant chapters (Chapter 15: Oscillations for Class 11, and Chapter 14: Oscillations for Class 12) in the NCERT Physics textbooks, and consistently practicing numerical problems will enhance your understanding of periodic motion and prepare you for the JEE exam.