Oscillations — Class 11 Physics Notes

Topics covered in Oscillations

1. 1.Introduction of Oscillation

   Short Answer

   Oscillations refer to the repeated back-and-forth movement of something between two positions or states. It's commonly seen in pendulums, springs, and waves.

   Long Answer

   Introduction to Oscillations:

   1. Definition: Oscillations are movements that repeat regularly over time, like a swing moving back and forth.

   2. Types of Oscillations:

      • Simple Harmonic Motion (SHM): The most fundamental type, where the force is directly proportional to the displacement and acts in the opposite direction.
      • Damped Oscillation: When the amplitude of oscillation decreases over time, usually due to friction or resistance.
      • Forced Oscillation: Occurs when an external force drives the oscillation, such as pushing a child on a swing.

   3. Characteristics:

      • Amplitude: Maximum distance from the equilibrium position.
      • Period: Time taken for one complete cycle of oscillation.
      • Frequency: Number of oscillations per unit time.

   4. Mathematical Expression: For a simple harmonic oscillator, =cos⁡(+)x(t)=Acos(ωt+ϕ), where x(t) is the displacement, A is the amplitude, ω is the angular frequency, and ϕ is the phase constant.

   5. Real-Life Applications:

      • In clocks (pendulum), music (vibrations of strings and air), and electronics (oscillators in circuits).
      • In understanding seismic waves and designing structures to withstand earthquakes.

   6. Career Relevance: Essential for careers in physics, engineering, music, and electronics.

2. 2.Periodic and Oscillatory Motions

   Short Answer

   Periodic motion is any motion that repeats at regular time intervals, while oscillatory motion is a type of periodic motion where an object moves back and forth within a certain range.

   Long Answer

   Periodic Motion:

   1. Definition: Motion that repeats itself after equal intervals of time, like the orbiting of planets.
   2. Characteristics: Regularity and predictability in time intervals; the motion recurs over equal time periods.
   3. Examples: The revolution of Earth around the sun, a swinging pendulum, the vibration of a guitar string.

   Oscillatory Motion:

   1. Definition: A specific type of periodic motion where an object moves back and forth over a central position.
   2. Characteristics: Includes amplitude (maximum displacement), frequency (number of oscillations per unit time), and period (time for one complete oscillation).
   3. Examples: A child on a swing, a mass attached to a spring, sound waves.
   4. Mathematical Expression: Often described by simple harmonic motion equations, such as =cos⁡(+)x(t)=Acos(ωt+ϕ).

   Key Differences:

   • All oscillatory motions are periodic, but not all periodic motions are oscillatory.
   • Oscillatory motion is restricted to movements around a central point, while periodic motion can include broader movements like rotations and revolutions.

3. 3.Period and frequency

   Short Answer

   Period is the time taken for one complete cycle of a periodic motion. Frequency is the number of cycles of a periodic motion that occur in one second.

   Long Answer

   Period (T):

   1. Definition: The period is the duration of time it takes for a wave, oscillation, or any other periodic motion to complete one full cycle.
   2. Measurement: Measured in seconds (s).
   3. Example: In a pendulum, the period is the time it takes to swing from one side to the other and back again.

   Frequency (f):

   1. Definition: Frequency is the number of complete cycles of periodic motion that occur in a unit of time, typically one second.
   2. Measurement: Measured in hertz (Hz), where 1 Hz = 1 cycle per second.
   3. Example: If a pendulum swings back and forth 2 times in 1 second, its frequency is 2 Hz.

   Relationship between Period and Frequency:

   • Period and frequency are inversely related. The formula connecting them is =1f=T1​, where f is the frequency and T is the period.
   • A shorter period means a higher frequency, and vice versa.

4. 4.Displacement

   Short Answer

   Displacement is the change in position of an object. It is a vector quantity, meaning it has both magnitude and direction.

   Long Answer

   1. Definition: Displacement is a vector quantity that represents the change in position of an object. It's not just the distance an object travels, but the overall change in position from the starting point to the end point.

   2. Formula and Derivation:

      • Consider an object moving from point A to point B.
      • If the coordinates of A are (1,1)(x1​,y1​) and B are (2,2)(x2​,y2​), then the displacement vector ⃗d can be represented as ⃗=⃗−⃗d=B−A.
      • This gives us ⃗=(2−1,2−1)d=(x2​−x1​,y2​−y1​).
      • The magnitude of the displacement vector can be found using the Pythagorean theorem: ∣⃗∣=(2−1)2+(2−1)2∣d∣=(x2​−x1​)2+(y2​−y1​)2​.

   3. Characteristics:

      • Has both magnitude and direction.
      • Is different from distance, which is scalar and only measures the length of the path traveled.

   4. Real-Life Example: If you walk from your home to school and then return home, your total displacement is zero because you end up at your starting point, even though you have traveled a distance.

   5. Use in Physics: Displacement is crucial in understanding motion, forces, and energy. It's used in equations of motion and in the study of vectors.

5. 5.Simple Harmonic Motion

   Short Answer

   Simple Harmonic Motion (SHM) is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction.

   Long Answer

   Simple Harmonic Motion (SHM):

   1. Definition: SHM is a type of oscillatory motion characterized by the property that the restoring force acting on the object is directly proportional to its displacement from its equilibrium position and is always directed towards that equilibrium position.

   2. Mathematical Expression:

      • The force F acting on an object in SHM is given by =−F=−kx, where k is the spring constant, and x is the displacement.
      • The motion can be described by =cos⁡(+)x(t)=Acos(ωt+ϕ), where A is the amplitude, ω is the angular frequency, t is time, and ϕ is the phase constant.

   3. Characteristics:

      • Periodic: The motion repeats after a regular interval.
      • Isochronous: The period of oscillation is constant and independent of amplitude for small displacements.
      • Energy Conservation: Energy in SHM oscillates between kinetic and potential, with total energy remaining constant.

   4. Examples:

      • A mass attached to a spring.
      • A pendulum with small angular displacement.

   5. Applications:

      • Used in watches (quartz oscillators).
      • In physics, to model systems like molecular vibrations and sound waves.

   6. Importance in Physics: SHM provides a foundational model for understanding more complex forms of periodic motion and wave phenomena.

6. 6.Simple Harmonic Motion and Uniform Circular Motion

   Short Answer

   Simple Harmonic Motion (SHM) and Uniform Circular Motion (UCM) are both periodic motions, but SHM occurs in a linear path while UCM occurs in a circular path. In UCM, an object moves at a constant speed along a circular path, whereas in SHM, the object oscillates back and forth along a straight line.

   Long Answer

   Simple Harmonic Motion (SHM):

   1. Path: Linear, with oscillations back and forth.
   2. Force: The restoring force is directly proportional to the displacement and acts in the opposite direction.
   3. Energy: In SHM, kinetic and potential energy are continuously exchanged, but total mechanical energy remains constant.
   4. Examples: Mass-spring system, pendulum (for small displacements).

   Uniform Circular Motion (UCM):

   1. Path: Circular, with constant speed along the circumference.
   2. Force: Centripetal force acts towards the center of the circle, keeping the object in circular motion.
   3. Energy: Kinetic energy remains constant if the speed is constant; no potential energy changes are involved.
   4. Examples: Motion of a satellite around a planet, a car turning around a circular track.

   Connection between SHM and UCM:

   • Projection: The projection of an object in UCM on a diameter of the circular path exhibits SHM. This means, if you look at the shadow of an object in UCM on a flat surface, that shadow will move with SHM.
   • Mathematics: The mathematics of SHM can be derived from the circular motion equations by considering the projection.

   Key Differences:

   • Motion Path: SHM is linear, while UCM is circular.
   • Force Nature: In SHM, the force is a restoring force, proportional to displacement. In UCM, it's the centripetal force, constant in magnitude and directed towards the center.
   • Energy Distribution: SHM involves the continuous transformation between kinetic and potential energy, whereas in UCM under constant speed, kinetic energy remains constant.

7. 7.Velocity and Acceleration in Simple Harmonic Motion

   Short Answer

   In Simple Harmonic Motion (SHM), velocity is the rate of change of displacement and acceleration is the rate of change of velocity. Velocity changes continuously, being maximum at the equilibrium position and zero at the extremes. Acceleration is always directed towards the equilibrium position and is maximum at the extremes.

   Long Answer

   Velocity in SHM:

   1. Description: Velocity in SHM varies sinusoidally. It's highest at the equilibrium point and zero at the maximum displacement points (amplitude).
   2. Mathematical Expression: If =cos⁡(+)x(t)=Acos(ωt+ϕ) is the displacement, the velocity v(t) is the first derivative of displacement with respect to time, given by =−sin⁡(+)v(t)=−Aωsin(ωt+ϕ).

   Acceleration in SHM:

   1. Description: Acceleration in SHM is also sinusoidal, always directed towards the equilibrium position. It's zero at the equilibrium point and maximum at the points of maximum displacement.
   2. Mathematical Expression: The acceleration a(t) is the derivative of velocity, or the second derivative of displacement, given by =−2cos⁡(+)a(t)=−Aω2cos(ωt+ϕ). It can also be written as =−2a(t)=−ω2x(t), showing that acceleration is proportional to the negative of displacement.

   Key Points:

   • Phase Difference: Velocity and acceleration are out of phase with displacement in SHM. When displacement is maximum, velocity is zero and acceleration is maximum in the opposite direction.
   • Amplitude Relation: Velocity is zero when the displacement is at its amplitude, and acceleration is maximum at this point.

8. 8.Force Law for Simple Harmonic Motion

   Short Answer
   

   Force law for simple harmonic motion (SHM) states that the force acting on an object in SHM is directly proportional to its displacement from the mean position and acts in the opposite direction. Mathematically, F = -kx, where F is the force, k is the spring constant, and x is the displacement.

   Long Answer Simple Harmonic Motion (SHM): It's a type of periodic motion where the force acting on the object is always directed towards a fixed point (equilibrium position) and is proportional to the displacement from that point.

   1. Force Law in SHM: The formula F = -kx describes this law.

      • F: Force acting on the object.
      • k: Spring constant, a measure of the stiffness of the spring.
      • x: Displacement from the mean position (equilibrium).

   2. Explanation:

      • Direct Proportionality: The force is directly proportional to the displacement. The farther the object from the equilibrium, the stronger the force pulling it back.
      • Negative Sign: Indicates the force acts in the opposite direction of the displacement.

   3. Real-life Example: Think of a pendulum or a spring with a weight attached. When you pull it and release, it moves back and forth. This motion is SHM, and the force that brings it back each time is described by F = -kx.

   4. Applications and Careers:

      • Engineering: Design of suspension systems in cars, watches, seismology equipment.
      • Physics & Research: Understanding wave motion, sound, and light properties.
      • Medicine: Designing equipment like MRI machines that use principles of SHM.

   5. Activity: Take a simple spring and attach a small weight to it. Stretch the spring and release. Observe how it moves back and forth, demonstrating SHM.

9. 9.Energy in Simple Harmonic Motion

   Short Answer

   In simple harmonic motion (SHM), energy is constantly exchanged between potential energy and kinetic energy, but the total energy remains constant. At maximum displacement, energy is all potential, and at the equilibrium position, it's all kinetic.

   Long Answer

   1. Concept of Energy in SHM: Energy in SHM is a dynamic interplay between kinetic energy (KE) and potential energy (PE).

   2. Kinetic Energy (KE):

      • KE = 12221​mv2, where m is mass, v is velocity.
      • Maximum at equilibrium (center point) where velocity is highest.

   3. Potential Energy (PE):

      • PE = 12221​kx2, where k is the spring constant, x is displacement.
      • Maximum at maximum displacement where the spring or pendulum is stretched or compressed the most.

   4. Conservation of Energy:

      • Total Energy (TE) = KE + PE remains constant.
      • Energy shifts from KE to PE and back as the object moves.

   5. Real-Life Example: A swinging pendulum. At the highest points, it's all PE. As it passes through the bottom, it's all KE.

   6. Applications and Careers:

      • Engineering: Designing clocks, oscillators in electronics.
      • Physics: Studying wave motion, sound.
      • Amusement Parks: Designing pendulum rides.

   7. Activity: Swing a pendulum or bounce a spring toy. Notice how the speed and stretching change, representing KE and PE changes.

10. 10.The Simple Pendulum

    
    

    Short Answer

    A simple pendulum consists of a small mass (called the bob) suspended from a fixed point by a string or rod. It demonstrates simple harmonic motion when displaced from its equilibrium position and released.

    Long Answer

    1. Components of a Simple Pendulum:

       • Bob: A small object of mass 'm', usually a metal ball.
       • String or Rod: Suspends the bob and is attached to a fixed point. Ideally weightless and inextensible.

    2. Motion Description:

       • When the bob is displaced from its resting position and released, it swings back and forth about the equilibrium position.
       • This motion is periodic and an example of simple harmonic motion (SHM).

    3. Period of a Simple Pendulum:

       • The time for one complete cycle (back and forth) is called the period.
       • Period, =2T=2πgl​​, where l is the length of the string and g is the acceleration due to gravity.

    4. Factors Affecting Period:

       • Only the length of the string and gravity affect the period.
       • Mass of the bob and amplitude of swing (provided it's small) do not affect the period.

    5. Real-Life Application and Careers:

       • Timekeeping: Historical clocks, like grandfather clocks.
       • Science: Studying gravitational acceleration.
       • Engineering: Designing sensors and oscillators.

    6. Activity:

       • Create a simple pendulum using a string and a small weight. Measure how the period changes with different string lengths.