Untitled Notes

Organization and Mission - New Jersey Center for Teaching & Learning (NJCTL) is dedicated to empowering teachers for school improvement. - Supported by the New Jersey Education Association (NJEA).

Date & Source: Presented on February 9, 2024, available at www.njctl.org
Table of Contents: - Simple Harmonic Motion (SHM) - SHM and Uniform Circular Motion (UCM) - Mass - Spring System Motion - Simple Pendulum - Mass - Spring System Energy - Sinusoidal Nature of SHM

Introduction to the Connection between SHM and UCM - SHM can be considered a one-dimensional projection of UCM. This foundational understanding allows existing concepts from UCM to be applied to SHM.

Definition of UCM: - UCM occurs when an object moves in a circle under the influence of a force directed towards the circle's center. - The speed remains constant, but the velocity changes as the direction varies.
Period (T): - Defined as the time required for one complete trip around the circular path. - Measured in seconds (s).
Calculating Period: - When time (t) is known for multiple revolutions (n), the period can be calculated as: $T = \frac{t}{n}$

Definition of Frequency: - Frequency refers to the number of revolutions completed in a specific time. - It is measured in hertz (Hz), which is equivalent to cycles per second (s^{-1}).
Calculating Frequency: - If time (t) for multiple revolutions (n) is known: $f = \frac{n}{t}$ - The formula inversely relates frequency and period: $f = \frac{1}{T}$

Velocity Formulas: - Magnitude of velocity is constant: $v = \frac{2 ext{π}r}{T}$ - In terms of frequency: $v = 2 ext{π}rf$

In UCM, an object completes one circle every T seconds and returns to the starting position.
In SHM, an object also returns to the starting position in T seconds.
Both types of motion are considered periodic.

A UCM green ball moves in two dimensions while an SHM red block oscillates in one dimension, always retaining the same height.
This shows that SHM acts as a one-dimensional projection of UCM with constant horizontal speed.

Conditions for SHM: - An oscillating object must be subjected to a restoring force that is proportional to and opposite its displacement from an equilibrium position, as defined by Hooke's Law. - The object experiences two forms of energy (kinetic and elastic potential energy), with energy continually transitioning between the two while conserving total energy.

Example 1: Period & Frequency of a Mass-Spring System - 40 complete oscillations in 8.0 seconds. - Given: n = 40, t = 8.0 s. - Calculate period: $T = \frac{t}{n} = \frac{8.0 s}{40} = 0.20 s$ - Calculate frequency: $f = \frac{n}{t} = \frac{40}{8.0 s} = 5.0 Hz$
Example 2: Frequency of a Simple Pendulum - Given: T = 2.0 s. - Frequency calculation: $f = \frac{1}{T} = \frac{1}{2.0 s} = 0.50 Hz$
Example 3: Period from Frequency - Given: f = 25 Hz. - Calculate period: $T = \frac{1}{f} = \frac{1}{25.0 Hz} = 0.04 s$

Description: A mass on a spring oscillates around an equilibrium point, capable of being in a state of compression or extension.
The motion follows the same principles established for SHM and UCM.
Energy Representation: - Kinetic energy (KE) and elastic potential energy (EPE) are calculated as follows: - $ext{EPE} = \frac{1}{2}kx^2$ where k is the spring constant and x is the displacement from equilibrium. - $ext{KE} = \frac{1}{2}mv^2$ where m is mass and v is the velocity.

Example 1: Amplitude and Period: - The mass-spring system with T = 2.0 s and A = 0.40 m travels in 4.0 s: - Maximum travel distance in one period: $4A = 4(0.40 m) = 1.6 m$ - In 4.0 s, it covers a distance of $2(1.6 m) = 3.2 m$ .
Example 2: Spring Force: - Calculate spring force for k = 100 N/m and x = 0.10 m. - Force is given by Hooke's Law: $F = -kx = -100 N/m imes 0.10 m = -10 N$

Energy in a mass-spring system consists of EPE and KE, with the total mechanical energy remaining constant throughout the oscillatory motion.

Energy states at various points in oscillation: - All EPE at maximum amplitudes: $E = \frac{1}{2}kA^2$ - All KE at equilibrium: $E = \frac{1}{2}mv_{max}^2$ where v_max is the maximum speed.

Description: A simple pendulum consists of a mass at the end of a negligible mass cord. The restoring force is gravity.
Energy Types: Kinetic and Gravitational Potential Energy are interchanged, with total energy constant.
In small angle approximations, the motion follows SHM properties, with the restoring force being proportional to displacement.

Given: L = 2.00 m (length), g = 9.8 m/s^2 (acceleration due to gravity).
Calculation of period: - $T = 2 ext{π} imes ext{sqrt}\frac{L}{g}$ - Substitute values: $T = 2 ext{π} imes ext{sqrt}\frac{2.00 m}{9.8 m/s^2}$

Position as a Function of Time: - General form is given by: $x(t) = A imes ext{cos}(\frac{2 ext{π}}{T}t)$ - Where A is the amplitude.
Velocity and Acceleration Functions: - Velocity: $v(t) = -A imes \frac{2 ext{π}}{T} imes ext{sin}(\frac{2 ext{π}}{T}t)$ - Acceleration: $a(t) = -\frac{A(2 ext{π})^2}{T^2} imes ext{cos}(\frac{2 ext{π}}{T}t)$

Graphs demonstrate the relationships of position, velocity, and acceleration over time: - At maximum displacement, velocity = 0, and acceleration = maximum. - As the mass passes through the equilibrium position, it has maximum velocity and zero acceleration.
The energy involves oscillating between kinetic and potential energy (graphs available in video resources).

The information presented encompasses all necessary details to understand the principles of Simple Harmonic Motion, relationships to Uniform Circular Motion, and associated mathematical calculations.
Use interactive resources to further reinforce concepts and practice with examples.