# Simple Harmonic Motion

Simple harmonic motion (SHM) is motion in which a restoring force is directly proportional to the displacement of an object

Nature's response to a perturbation or disturbance is often SHM

# Circular Motion vs. SHM

# Position, Velocity, Acceleration

# Frequency and Period

Frequency

Frequency is the number of revolutions or cycles which occur each second

Symbol is f

Units are 1/s, or Hertz (Hz)

Period

Period is the time it takes for one complete revolution, or cycle.

Symbol is T

Unites are seconds (s)

T = time for 1 cycle = time for 1 revolution

Relationship

# Angular Frequency

Angular frequency is the number of radians per second, and it corresponds to the angular velocity for an object traveling in uniform circular motion

Relationship

# Example 1: Oscillating System

An oscillating system is created by a releasing an object from a maximum displacement of 0.2 meters. The object makes 60 complete oscillations in one minute

Determine the object's angular frequency

What is the object's position at time t=10s?

At what time is the object at x=0.1m?

# Mass on a Spring

# Example 2: Analysis of Spring-Block System

A 5-kg block is attached to a 2000 N/m spring as shown and displaced a distance of 8 cm from its equilibrium position before being released.

Determine the period of oscillation, the frequency, and the angular frequency for the block

# General Form of SHM

# Graphing SHM

# Energy of SHM

When an object undergoes SHM, kinetic and potential energy both vary with time, although total energy (E=K+U) remains constant

# Horizontal Spring Oscillator

# Vertical Spring Oscillator

# Springs in Series

# Springs in Parallel

# The Pendulum

A mass m is attached to a light string that swings without friction about the vertical equilibrium position

# Energy and the Simple Pendulum

# Frequency and Period of a Pendulum

# Period of a Physical Pendulum

# Example 3: Deriving Period of a Simple Pendulum

# Example 4: Deriving Period of a Physical Pendulum

# Example 5: Summary of Spring-Block System

# Example 6: Harmonic Oscillator Analysis

A 2-kg block is attacked to a spring. A force of 20 N stretches the spring to a displacement of 0.5 meter

The spring constant

The total energy

The speed at the equilibrium position

The speed at x=0.30 meters

The speed at x=-0.4 meters

The acceleration at the equilibrium position

The magnitude of the acceleration at x=0.5 meters.

The net force at equilibrium position

The net force at x=0.25 meter

Where does kinetic energy = potential energy

# Example 7: Vertical Spring Block Oscillator

A 2-kg block attached to an un-stretched spring of spring constant k=200 N/m as shown in the diagram below is released from rest. Determine the period of the block's oscillation and the maximum displacement of the block from its equilibrium while undergoing simple harmonic motion.

# 2009 Free Response Question 2

# 2010 Free Response Question 3