Measuring Simple
Harmonic Motion
Chapter 12 Section 2
Measuring The Motion

There are 3 things that determine the motion
of a mass in simple harmonic motion.



Amplitude
Period
Frequency
Amplitude


Amplitude – The maximum displacement
from the equilibrium position.
Can be measured in different ways


Pendulum – The angle (radians) between the
equilibrium position and the maximum
displacement.
Spring-mass – The maximum amount (meters)
stretched or compressed from the equilibrium
position.
Period


Period – The time it takes to execute a
complete cycle of motion.
For example:



If it takes 5 seconds for a person on a swing to
swing back and forth, then the period of the
motion would be 5 seconds.
SI units for period – Seconds (s)
Variable given for period – Capital letter (T)
Displacement For Period

The displacement of an object in simple
harmonic motion during the time of 1T (time
to compete one cycle) is “ZERO.”
Frequency


Frequency – The number of cycles or
vibrations per unit time.
For example:

The person on the swing completes one cycle in 5
seconds, the frequency would be 1/5 cycles per
second or 0.2 cycles per second.
Units For Frequency

SI units for frequency – S-1



Variable for frequency – lower case letter (f)
In the case of the person swinging, the
frequency would be:


This is known as Hertz (Hz)
0.2 cycles per second = 0.2 Hz
A typical TV set has a frequency of 60Hz,
which means 60 frames per second.
Differences Between Period
and Frequency



Period is time per cycle.
Frequency is the number of cycles per unit
time.
They are inversely proportional.
Equations For Frequency and
Period

1
𝑓=
𝑇
1
𝑇=
𝑓
If the period or the frequency is known, this
relationship can be used to calculate the
other value.


Period (s)
Frequency (Hz)
Determining The Period of a
Pendulum


The strings length and the free fall
acceleration determine the period of a simple
pendulum.
Things that don’t determine the Period:


Amplitude (for angles less then 15 degrees)
Mass of the bob
Simple Pendulum Equation
𝑙
𝑇 = 2𝜋
𝑔



𝑇 = 𝑃𝑒𝑟𝑖𝑜𝑑 𝑠
𝑙 = 𝐿𝑒𝑛𝑔𝑡ℎ 𝑚
𝑚
𝑔 = 𝐴𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑑𝑢𝑒 𝑡𝑜 𝐺𝑟𝑎𝑣𝑖𝑡𝑦 (9.8𝑠2)
Example Problem #1

A desktop toy swings back and forth once
every 1.0 seconds. How tall is this toy?
Example Problem #1 Answer

length = 0.25m
Example Problem #2

What is the period of a 3.98m long
pendulum?
Example Problem #2 Answer

T = 4.00 seconds
Period of a Mass-Spring
System


The mass attached to the spring and the
spring constant (k) determine the period.
Things that don’t determine the period:

Amplitude
Period of a Mass-Spring
System Equation
𝑚
𝑇 = 2𝜋
𝑘

𝑇 = 𝑃𝑒𝑟𝑖𝑜𝑑 𝑠
𝑚 = 𝑀𝑎𝑠𝑠 𝑘𝑔

𝑘 = 𝑆𝑝𝑟𝑖𝑛𝑔 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝑁
𝑚
Example Problem #3

A 1.0 kg mass attached to one end of a
spring completes one oscillation every 2.0
seconds. Find the spring constant.
Example Problem #3 Answer

k = 9.9N/m