Chapter 14 - Oscillations

14.1 Oscillations of a Spring

Oscillatory motion occurs when a mass attached to a spring moves back and forth about an equilibrium position, provided there is no friction. This is the most basic type of oscillatory motion, known as Simple Harmonic Motion (SHM).

• Restoring Force: The force exerted by the spring is proportional to the displacement from equilibrium: .

• Positive Displacement: When the spring is stretched, , and the force acts to restore the mass to equilibrium.

• Negative Displacement: When the spring is compressed, , and the force again acts to restore equilibrium.

• Differential Equation: Applying Newton's second law yields:

• Natural Angular Frequency: Defined as .

• General Solution: The position as a function of time is: or

Additional info: The solution can also be expressed as a sum of sine and cosine functions, depending on initial conditions.

14.2 Simple Harmonic Motion (SHM)

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

• General Solution: , where:

  • A: Amplitude (maximum displacement)

  • : Phase (determined by initial conditions)

  • : Angular frequency

• Alternative Forms:

  • , where

• Properties of SHM:

  • Periodic motion described by sine or cosine functions

  • Characterized by amplitude, period, and phase

  • Acceleration is proportional and opposite to displacement:

Example 1: Determining velocity and acceleration signs from a position-time graph. Positive velocity and negative acceleration occur where the slope is positive and the position is negative.

Properties of SHM

• Displacement: Measured from the equilibrium point.

• Amplitude: Maximum displacement from equilibrium.

• Cycle: One complete to-and-fro motion.

• Period (T): Time for one cycle (unit: seconds).

• Frequency (f): Number of cycles per second (unit: Hertz, Hz).

• Equations:

• Period Independence: The period is independent of amplitude for SHM.

Velocity and Acceleration in SHM

• Velocity:

• Acceleration:

• Maximum Values:

Example 2: Vertical Spring Oscillation

• Given: g, m, s

• Spring Constant: N/m

• Equations:

• Maximum Velocity: m/s

• Maximum Acceleration: m/s2

Example 3: Addition of Sinusoidal Waves

• Sum of Sines:

• Application: , where ,

• Result: The sum is another sine or cosine with the same frequency.

Example 4: Determining Amplitude, Period, and Frequency from Graphs

• Given position-time graphs, identify amplitude (peak value), period (time for one cycle), and frequency ().

• Write equations in sine and cosine form: and .

14.3 Energy in the Simple Harmonic Oscillator

The total mechanical energy in SHM is constant, as only conservative forces are present.

• Potential Energy:

• Kinetic Energy:

• Total Energy:

• Energy Exchange: Energy oscillates between kinetic and potential forms during SHM.

Application to Molecules: For small displacements, the potential energy curve of a hydrogen molecule can be approximated as quadratic, leading to SHM for atomic vibrations.

14.4 SHM Related to Uniform Circular Motion

Simple harmonic motion can be visualized as the projection of uniform circular motion onto one axis.

• Velocity Component:

• Equivalence: The x-component of velocity in circular motion matches the velocity in SHM.

14.5 The Simple Pendulum

A simple pendulum consists of a point mass suspended from a massless, taut string or rod, swinging under gravity in a vertical plane.

• Forces:

  • Tension in the string (constrains motion to an arc)

  • Weight (acts downward)

• Arc Length: , where is the length and is the angular displacement.

• Equations of Motion:

  • Velocity:

  • Acceleration:

  • Restoring force:

  • Newton's law:

  • For small angles:

  • Differential equation:

  • Angular frequency:

• Energy:

  • Kinetic:

  • Potential:

  • Total:

Oscillator Properties: Comparison Table

Frequency and Angular Frequency

• Spring-Mass System:

• Pendulum:

• Natural/Resonant Frequency: The frequency at which a system oscillates in the absence of driving or damping forces.

Summary of Chapter 14

• For SHM, the restoring force is proportional to displacement.

• The period is the time for one cycle; frequency is cycles per second.

• Energy in SHM oscillates between kinetic and potential forms, but total mechanical energy remains constant.