Bihar Board - Class 11 physics - Chapter 14: Oscillations Handwritten Notes

The chapter "Oscillations" explores the periodic motion of objects and the physical principles governing them. Oscillatory motion is fundamental in nature and is observed in systems like pendulums, springs, and even atomic vibrations. This chapter introduces key concepts such as simple harmonic motion (SHM), damping, resonance, and the energy associated with oscillatory systems.

Key Points

1. Oscillatory Motion

   • Oscillatory motion is repetitive motion around a central point or equilibrium position. Examples include the swinging of a pendulum and the vibrations of a spring.
   • Types of oscillations include free oscillations, damped oscillations, and forced oscillations.

2. Simple Harmonic Motion (SHM)

   • SHM is a special type of oscillatory motion where the restoring force is directly proportional to the displacement and acts in the opposite direction. $$F = -kx$$where $F$ is the restoring force, $k$ is the spring constant, and $x$ is the displacement.
   • The displacement as a function of time is given by: $$x(t) = A \cos(\omega t + \phi)$$ where $A$ is the amplitude, $\omega$ is the angular frequency, $t$ is time, and $\phi$ is the phase constant.

3. Time Period and Frequency

   • Time Period (T): The time taken for one complete oscillation. $$T = \frac{2\pi}{\omega}$$
   • Frequency (f): The number of oscillations per unit time. $$f = \frac{1}{T}$$
   • Angular Frequency ($\omega$): The rate of change of angular displacement with time. $$\omega = 2\pi f$$
     

4. Energy in SHM

   • The total energy in SHM is conserved and alternates between kinetic and potential energy.
     • Potential Energy: $$U = \frac{1}{2}kx^2$$
       
     • Kinetic Energy: $$K = \frac{1}{2}m\omega^2(A^2 - x^2)$$
       
     • Total Energy: $$E = \frac{1}{2}kA^2$$
       

5. Damped Oscillations

   • In real-world systems, oscillations gradually decrease in amplitude over time due to resistive forces like friction or air resistance.
   • The damping force is proportional to the velocity: $$F_{\text{damping}} = -bv$$ where $b$ is the damping coefficient.

6. Forced Oscillations and Resonance

   • Forced Oscillations: Occur when an external periodic force drives the system.
     • Equation: $$m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = F_0 \cos(\omega t)$$
       
   • Resonance: A phenomenon where the system oscillates with maximum amplitude when the driving frequency matches the natural frequency.

7. Pendulum Motion

   • A simple pendulum exhibits SHM for small angular displacements.
     • Time period of a simple pendulum: $$T = 2\pi \sqrt{\frac{L}{g}}$$ where $L$ is the length of the pendulum and $g$ is the acceleration due to gravity.

8. Applications of Oscillations

   • Oscillatory motion is crucial in designing clocks, musical instruments, electrical circuits (LC oscillations), and even quantum systems.

Conclusion

The chapter "Oscillations" provides a comprehensive understanding of periodic motion, its characteristics, and its mathematical representation. By exploring SHM, damped motion, and resonance, students gain insights into the fundamental principles that govern oscillatory systems in both natural and artificial environments.

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