Conservation of Energy in Simple Harmonic Motion and Introduction to Waves

Conservation of Energy in Simple Harmonic Motion

Elastic Potential Energy in a Spring

When a mass is attached to a spring, the system can store energy in the form of elastic potential energy. The potential energy stored in a spring compressed or stretched by a distance x from equilibrium is given by:

• Formula: $PE_{spring} = \frac{1}{2}kx^2$

• k: Spring constant (N/m)

• x: Displacement from equilibrium (m)

Energy Transformations in a Mass-Spring System

As a block moves toward a spring, its energy transforms between kinetic and potential forms, but the total mechanical energy remains constant (assuming no friction).

• Initial State: All energy is kinetic as the block moves with velocity $v_i$ toward the spring.

• Intermediate State: The block compresses the spring by distance $x$ and slows down. Energy is shared between kinetic and elastic potential energy.

• Maximum Compression: The block stops momentarily at maximum compression $x_{max}$; all energy is stored as elastic potential energy.

• Conservation of Energy Equation:

$E_{tot} = KE + PE_{spring} = \frac{1}{2}mv^2 + \frac{1}{2}kx^2 = \text{constant}$

Energy Diagrams and Turning Points

The potential energy curve for a spring is a U-shaped parabola. At the turning points ($x = \pm A$), all energy is potential; at equilibrium ($x = 0$), all energy is kinetic.

• Turning Points: Maximum displacement, zero velocity, maximum potential energy.

• Equilibrium: Zero displacement, maximum velocity, maximum kinetic energy.

Simple Harmonic Motion: Equations and Parameters

Equations of Motion

The position, velocity, and acceleration of a mass on a spring in simple harmonic motion (SHM) are described by sinusoidal functions:

• Position: $x(t) = A \sin\left(2\pi \frac{t}{T} + \phi\right)$

• Velocity: $v(t) = \frac{dx}{dt} = A \frac{2\pi}{T} \cos\left(2\pi \frac{t}{T} + \phi\right)$

• Acceleration: $a(t) = \frac{dv}{dt} = -A \left(\frac{2\pi}{T}\right)^2 \sin\left(2\pi \frac{t}{T} + \phi\right)$

• Angular Frequency: $\omega = \frac{2\pi}{T}$

Where:

• A: Amplitude (maximum displacement)

• T: Period (time for one complete oscillation)

• \phi: Phase constant (determined by initial conditions)

Period of Oscillation

• Mass-Spring System: $T_{spring} = 2\pi \sqrt{\frac{m}{k}}$

• Pendulum: $T_{pend} = 2\pi \sqrt{\frac{L}{g}}$

• Frequency: $f = \frac{1}{T}$

Note: The period of a mass-spring system depends only on the mass and spring constant, not on gravity. The period of a pendulum depends on the length and the local gravitational acceleration.

Waves: Types and Properties

Nature of Waves

A wave is a disturbance that travels through a medium (solid, liquid, or gas), transferring energy without transferring matter. The frequency of a wave is the number of vibrations per second.

Types of Waves

• Transverse Waves: The displacement of the medium is perpendicular to the direction of wave travel.

• Longitudinal Waves: The displacement of the medium is parallel to the direction of wave travel.

• Comparison:

Type	Direction of Displacement	Example
Transverse	Perpendicular to wave direction	Waves on a string
Longitudinal	Parallel to wave direction	Sound waves

Wave Properties

• Wavelength (\(\lambda\)): The distance between two consecutive points in phase (e.g., crest to crest).

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

• Period (T): Time for one cycle ($T = 1/f$).

• Wave Speed (v): $v = \lambda f$

General Form of a Traveling Wave

The displacement of a point on a string as a function of position and time is given by:

$y(x, t) = A \sin\left(2\pi \frac{x}{\lambda} \pm 2\pi \frac{t}{T} + \phi\right)$

• Amplitude (A): Maximum displacement

• Wave number (k): $k = \frac{2\pi}{\lambda}$

• Angular frequency (\(\omega\)): $\omega = 2\pi f$

• Phase constant (\(\phi\)): Sets the initial phase at $x=0$, $t=0$

• Direction: The sign in the argument determines the direction of travel (− for +x, + for −x)

Wave Graphs

• Snapshot Graph: Displacement as a function of position at a fixed time (shows the shape of the wave at an instant).

• History Graph: Displacement as a function of time at a fixed position (shows how a point in the medium moves over time).

Phase and Phase Difference

The phase of a wave at a point describes its position within the cycle. The phase difference between two points separated by $\Delta x$ is:

$\Delta \phi = 2\pi \frac{\Delta x}{\lambda}$

Wave Speed on a String

The speed of a wave on a stretched string depends on the tension ($F_T$) and the mass per unit length ($\mu$):

$v_{string} = \sqrt{\frac{F_T}{\mu}}$

• $F_T$: Tension in the string (N)

• $\mu$: Mass per unit length (kg/m)

Summary Table: Key Equations

Quantity	Equation	Description
Spring Force	$F_{spring} = -k \Delta x$	Hooke's Law
Elastic Potential Energy	$PE_{spring} = \frac{1}{2}kx^2$	Energy stored in a spring
Kinetic Energy	$KE = \frac{1}{2}mv^2$	Energy of motion
Total Mechanical Energy	$E_{tot} = KE + PE_{spring}$	Conserved in SHM
Period (Spring)	$T_{spring} = 2\pi \sqrt{\frac{m}{k}}$	Oscillation period for mass-spring
Period (Pendulum)	$T_{pend} = 2\pi \sqrt{\frac{L}{g}}$	Oscillation period for pendulum
Wave Speed	$v = \lambda f$	Speed of a wave
Wave on String	$v_{string} = \sqrt{\frac{F_T}{\mu}}$	Speed on a stretched string
Wave Function	$y(x, t) = A \sin(kx \pm \omega t + \phi)$	General wave equation

Additional info: The notes above integrate and expand upon the provided lecture slides, including definitions, equations, and academic context for simple harmonic motion and wave phenomena, as relevant to a college-level physics course.