Oscillations and Waves: Comprehensive NEET Physics Notes

1. Oscillations

1.1 Simple Harmonic Motion (SHM)

Formula:$x(t)=A\cos (\omega t+\varphi )$

• Explanation:
• • $x(t)$: Displacement as a function of time
  • $A$: Amplitude of motion
  • $\omega$: Angular frequency
  • $\varphi$: Phase constant
• Derivation:
• • The SHM equation is derived from the restoring force being proportional to the displacement, leading to the differential equation: $\frac{d^{2}x}{d{t}^2}+{\omega }^{2}x=0$
• Example Application:
• • Problem: A particle oscillates with an amplitude of 5 cm and a frequency of 2 Hz. Find its displacement after 0.5 seconds if the initial phase is zero.
  • Solution:
  • • Given: $A=5\text{cm},\nu =2\text{Hz},t=0.5\text{s},\varphi =0$
    • Find: $x(0.5)=A\cos (2\pi \nu t+\varphi )=5\cos (2\pi \times 2\times 0.5+0)=5\cos (2\pi )=5\text{cm}$

1.2 Velocity and Acceleration in SHM

Formula:

• Velocity: $v(t)=-\omega A\sin (\omega t+\varphi )$
• Acceleration: $a(t)=-{\omega }^{2}A\cos (\omega t+\varphi )$
• Explanation:
• • Velocity is the first derivative of displacement with respect to time.
  • Acceleration is the second derivative of displacement with respect to time.
• Example Application:
• • Problem: For the SHM described above, find the velocity and acceleration at $t=0.5\text{s}$.
  • Solution:
  • • Velocity: $v(0.5)=-2\pi \times 2\times 5\times \sin (2\pi \times 2\times 0.5)=0\text{cm/s}$
    • Acceleration: $a(0.5)=-4{\pi }^2\times 5\times \cos (2\pi \times 2\times 0.5)=-197.4{\text{cm/s}}^2$

1.3 Energy in SHM

Formula:

• Total Energy: $E=\frac{1}{2}k{A}^2$
• Kinetic Energy: $K=\frac{1}{2}m{v}^2=\frac{1}{2}k(A^2-x^2)$
• Potential Energy: $U=\frac{1}{2}k{x}^2$
• Explanation:
• • Total energy in SHM remains constant and is the sum of kinetic and potential energies.
• Example Application:
• • Problem: Calculate the kinetic and potential energy when the displacement is 3 cm for an SHM with $A=5\text{cm}$ and $k=50\text{N/m}$.
  • Solution:
  • • $U=\frac{1}{2}\times 50\times (0.03{)}^2=0.0225\text{J}$
    • $K=\frac{1}{2}\times 50\times ((0.05{)}^2-(0.03{)}^2)=0.0625-0.0225=0.04\text{J}$

2. Waves

2.1 Wave Equation

Formula: $y(x,t)=A\sin (kx-\omega t+\varphi )$

• Explanation:
• • $(x,t)$: Displacement of the wave at position $x$ and time $t$
  • $k$: Wave number, $k=\frac{2\pi }{\lambda }$
  • $\omega$: Angular frequency, $\omega =2\pi \nu$
• Example Application:
• • Problem: A wave traveling along a string is described by the equation $y(x,t)=0.02\sin (2\pi (30x-400t))$. Find the wavelength and speed of the wave.
  • Solution:
  • • Wavelength @$\lambda =\frac{2\pi }{k}=\frac{2\pi }{2\pi \times 30}=\frac{1}{30}\text{m}$
    • Speed $v=\frac{\omega }{k}=\frac{400}{30}=13.33\text{m/s}$

2.2 Speed of a Wave on a String

Formula: $v=\sqrt{\frac{T}{\mu }}$