### Notes from Toppers

**1. Oscillatory Motion:**

**Definition:** Oscillatory motion is a periodic motion where the body moves to and fro about a fixed point, called the mean position.

**Characteristics:**

- Oscillatory motion repeats itself at regular intervals.
- The body’s displacement from the mean position is a sinusoidal function of time.
- The body’s velocity and acceleration are also sinusoidal functions of time.
- The period of oscillation is the time taken for one complete oscillation.
- Frequency of oscillation is the number of oscillations per unit time.

**Reference:** NCERT Physics Part 1, Chapter 15, Oscillations.

**2. Displacement, Velocity, and Acceleration in SHM:**

**Equations:**

- $$Displacement, x=A\sin\omega t$$
- $$Velocity, v=A\omega \cos \omega t$$
- $$Acceleration, a=-A\omega^2 \sin\omega t$$ where,
- (A) is the amplitude of oscillation.
- (\omega) is the angular frequency of oscillation.

**Graphical Representation:**

(Displacement) vs. (Time) graph: [Image of a sine curve]

(Velocity versus Time) graph: [Image of a cosine curve]

(Acceleration) vs. (Time) graph: [Image of a sine curve, shifted down by (\pi/2)]

**Phase and Phase Difference:**

- The phase of an oscillation is the fraction of the way through a complete oscillation that has occurred.
- The phase difference between two oscillations is the difference in their phases.

**Reference:** NCERT Physics Part 1, Chapter 15, Oscillations.

**3. Period, Frequency, and Amplitude in SHM:**

**Definitions:**

**Period (T)**is the time taken for one complete oscillation.**Frequency (f)**is the number of oscillations per second.**Amplitude (A)**is the maximum displacement from the mean position.

**Relationship:**

- $$f=\frac{1}{T}$$
- (\omega=2\pi f) where (\omega) is the angular frequency.

**Reference:** NCERT Physics Part 1, Chapter 15, Oscillations.

**4. Energy in SHM:**

**Total Energy:** In SHM, the total energy is the sum of kinetic energy and potential energy:

$$E=K+U=\frac{1}{2}kA^2\cos^2\omega t+\frac{1}{2}kA^2\sin^2\omega t=\frac{1}{2}kA^2$$

- where (k) is the spring constant.

**Conservation of Energy:**
The total energy of a system in SHM remains constant.

**Reference:** NCERT Physics Part 1, Chapter 15, Oscillations.

**5. Applications of SHM:**

**Springs:**Springs oscillate when stretched or compressed. The period of oscillation of a spring is given by $$T=2\pi\sqrt{\frac{m}{k}}$$**Pendulums:**A pendulum is a weight suspended from a pivot point. The period of oscillation of a pendulum is given by $$T=2\pi\sqrt{\frac{l}{g}}$$ where (l) is the length of the pendulum and (g) is the acceleration due to gravity.**Oscillating Masses:**Masses attached to springs or suspended from pivots can oscillate. The period of oscillation depends on the mass and the spring constant or length of the pendulum.**Measuring Instruments:**SHM is used in various measuring instruments, such as accelerometers and seismometers.