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.