⏱️ Oscillations

1. Periodic and Oscillatory Motion

Periodic Motion: Motion that repeats itself at regular intervals (e.g., Earth around Sun).
Oscillatory (Vibratory) Motion: To-and-fro motion about a fixed mean position (e.g., pendulum). All oscillatory motions are periodic, but not all periodic motions are oscillatory.

2. Simple Harmonic Motion (SHM)

A special type of oscillatory motion where the restoring force is directly proportional to the displacement from the mean position and is always directed towards the mean position.

3. Kinematics of SHM

If displacement y(t) = A sin(ωt + φ), then:

• Velocity: v = dy/dt = Aω cos(ωt + φ) = ω√(A² - y²)
• Acceleration: a = dv/dt = -Aω² sin(ωt + φ) = -ω²y

Notice that acceleration is directly proportional to negative displacement, which is the defining property of SHM.

4. Energy in SHM

Potential Energy: U = ½ ky² = ½ mω²y²
Kinetic Energy: K = ½ mv² = ½ mω²(A² - y²)
Total Energy: E = K + U = ½ mω²A² (Constant, independent of time or position).

5. Simple Pendulum

Time period of a simple pendulum of length L is given by: T = 2π √(L/g).