Oscillations and Waves – JEE Mains Physics
1. Oscillations and Periodic Motion
- Oscillations are repetitive motions that occur in a fixed period. The time period (T) is the time taken to complete one full cycle of the oscillation, and the frequency (f) is the number of oscillations per unit time.
- Displacement as a function of time for periodic motion can be described using periodic functions like sine and cosine.
2. Simple Harmonic Motion (S.H.M.) and Its Equation
- Simple Harmonic Motion (S.H.M.) is a type of oscillatory motion where the restoring force is directly proportional to the displacement and acts in the opposite direction.
- The equation for S.H.M. is:
F = -kx
where F is the restoring force, k is the force constant, and x is the displacement.
3. Phase in Oscillations
- The phase of oscillation defines the position of the object in its cycle of motion at any given point in time.
4. Oscillations of a Spring: Restoring Force and Force Constant
- The restoring force for a spring is given by Hooke’s law:
F = -kx
where k is the spring constant and x is the displacement from equilibrium.
5. Energy in S.H.M.: Kinetic and Potential Energies
- In Simple Harmonic Motion, energy oscillates between kinetic and potential forms.
- The total mechanical energy (E) is constant:
E = (1/2)kA²
where A is the amplitude of oscillation. - The kinetic energy (K.E.) is maximum at the equilibrium position, and the potential energy (P.E.) is maximum at the extremes of the motion.
6. Simple Pendulum: Derivation of Expression for Its Time Period
- The time period (T) of a simple pendulum is derived as:
T = 2π√(L/g)
where L is the length of the pendulum and g is the acceleration due to gravity.
7. Wave Motion
- Wave motion refers to the propagation of disturbances through a medium. Waves can be classified as longitudinal or transverse.
8. Longitudinal and Transverse Waves
- Longitudinal waves: In longitudinal waves, the displacement of particles is parallel to the direction of wave propagation (e.g., sound waves).
- Transverse waves: In transverse waves, the displacement of particles is perpendicular to the direction of wave propagation (e.g., waves on a string).
9. Speed of the Travelling Wave
- The speed of a wave (v) is given by the relation:
v = fλ
where f is the frequency and λ is the wavelength of the wave.
10. Displacement Relation for a Progressive Wave
- The displacement for a progressive wave can be represented as:
y(x,t) = A sin(kx - ωt + φ)
where A is the amplitude, k is the wave number, ω is the angular frequency, and φ is the phase constant.
11. Principle of Superposition of Waves
- The principle of superposition states that when two or more waves overlap, the resultant displacement at any point is the algebraic sum of the displacements of the individual waves.
12. Reflection of Waves
- When waves encounter a boundary, they may reflect back. The reflection of waves follows specific rules, including the inversion of transverse waves.
13. Standing Waves in Strings and Organ Pipes
- Standing waves are formed when two waves of the same frequency and amplitude traveling in opposite directions interfere with each other.
- In a string fixed at both ends, standing waves form at specific frequencies called harmonics.
- In organ pipes, standing waves form inside the pipe, with the frequency depending on the pipe’s length and whether it is closed or open at the ends.
14. Fundamental Mode and Harmonics
- The fundamental mode is the lowest frequency at which a system oscillates. Harmonics are integer multiples of the fundamental frequency.
15. Beats
- Beats occur when two waves of slightly different frequencies interfere with each other, creating a variation in amplitude over time.
- The beat frequency is given by:
f_beat = |f₁ - f₂|
where f₁ and f₂ are the frequencies of the two waves.