# PHYS 2108 Lecture Notes - Lecture 8: Simple Harmonic Motion, Free Body Diagram, Stopwatch

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16 Jun 2018
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8-1
LAB 8:
Simple Harmonic Motion
Hookeâs Law
We all have some familiarity with springs. If we stretch a spring away from its relaxed (unstretched)
length, if exerts a pulling force to try to return to its shorter state. Likewise, if we compress the spring, it
exerts a pushing force to try to return to its lengthened state. This is called a restoring force: a force
opposing the change to restore the system to its relaxed state.
For most springs, the relationship between restoring force and displacement from the relaxed state is
linear. For example, if we double the stretching (or compression), the force doubles. This relationship is
called Hookeâs Law, and any spring obeying it is called an ideal spring.
îš
îŽ±î”î”îîîŽ±î
Eqn. 8-1
Hookeâs Laî
The negative sign is strictly a theoretical consideration to make the force vector in the opposite direction
of the displacement. In free body diagrams you only need to worry about the magnitude (because you
draw the vector in the correct direction). The coefficient k is called the spring constant and represents
the spîiî¶gâs stiffî¶ess. A stiffeî spring resists displacement more, so it has a higher k.
Consider the mass-spring system to the right. If it is in equilibrium,
the forces on it are balanced with equal magnitudes but opposite
directions. The downward force is of course just the weight of the
hanging mass. The spring counters this with its restoring force.
Hookeâs Laî depeî¶ds oî¶ us assuî”iî¶g the staîtiî¶g positioî¶ foî
displaîeî”eî¶t î”easuîeî”eî¶ts is at î = 0. If it is î¶ot, thatâs okaî. We
simply need to say the displacement is the difference in position
aî¶d î”odifî Hookeâs Laî as îš
îŽ±î”î”îïșîîŻî”îîŹŽï»
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îŽ±
.
Analyzing the free body diagram with a = 0,
îîî”îîî”îîœî”îČ
îîî”îî
Spring Potential Energy
Beîause îeâîe applîiî¶g a foîîe agaiî¶st the îestoîiî¶g foîîe through some displacement distance, îeâîe
doing work on the spring. That î”eaî¶s îeâîe stoîiî¶g poteî¶tial eî¶eîgî iî¶to it.
îČî§ î”îł
îŽîîîŹ¶î
Eqn. (8-2)
Spring Potential Energy
Real Springs
An ideal spîiî¶g oîeîs Hookeâs Laî liî¶eaîlî. A real spring is not as perfect. Most springs can be called
ideal îeîause theî oîeî Hookeâs Laî foî sî”all displaîeî”eî¶ts. At soî”e poiî¶t the appîoîiî”atioî¶ îîîeaks
doîî¶î if it stîetîhes too î”uîh. Iî¶ these îegioî¶s the spîiî¶g î”aî peîî”aî¶eî¶tlî defoîî” oî the î”etal î”ay
îeakeî¶. Otheî sîsteî”s oîeîiî¶g Hookeâs Laî îŸpeî¶dula, î”ostlîîż aîe liî¶eaî foî sî”all aî¶gles, îut îeîoî”e
îeîî diffeîeî¶t if theîâîe displaîed too faî.
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8-2
Simple Harmonic Motion
Aî¶î oîjeît that oîeîs Hookeâs Laî îill uî¶deîgo osîillatioî¶ if distuîîed fîoî” its eîuiliîîiuî” positioî¶. As
long as the restoring force is linear, the resulting oscillation is called simple harmonic motion.
The object must be excited to oscillate by giving it some energy. It can either be given kinetic energy at
its equilibrium position by striking it, or it can be given potential energy by being displaced away from its
equilibrium position. During the oscillation, the system is constantly trading its energy between KE and
PE forms.
Some key features are present in simple harmonic oscillation. The maximum displacement away from
equilibrium (and max PE, zero KE state) is called the amplitude (A) of oscillation. The amount of time for
the oscillator to make one complete cycle is called the period (T). The frequency (f) is how often the
osîillatoî îoî”pletes îîîles îŸkiî¶d of like a îspeedî iî¶ osîillatioî¶s peî seîond, or Hertz [Hz]).
The motion of a simple harmonic oscillator is described by
îî”îŁîîżîîïșîŽîšîîîï»
Tîig fuî¶îtioî¶s oî¶lî îoîk oî¶ aî¶gles, so the îaî¶gleî ïșîŽîšîîîï» must be considered to be in radians, not
degîees. This is due to îadiaî¶s îeiî¶g a îî¶atuîalî uî¶it îoî”posed of îatios.
If aî¶ osîillatoîâs fîeîueî¶îî is 3 osîillatioî¶s/s oî 3 Hz, for example, then each oscillation takes 1/3 of a
seîoî¶d to îoî”plete. Bî eîaî”ple îeâîe shoîed the peîiod of osîillatioî¶ aî¶d frequency are reciprocal of
one another.
îî”îł
î¶î
Eqn. 8-3
Period and Frequency
The frequency of oscillations depends on how easily the spring can move the attached mass. The
resulting period depends upon the systems parameters k and m as:
îł
îî”î¶î”îŽîšî¶§î
îî
Eqn. 8-4
Period for Ideal Spring-Mass
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## Document Summary

If we stretch a spring away from its relaxed (unstretched) length, if exerts a pulling force to try to return to its shorter state. Likewise, if we compress the spring, it exerts a pushing force to try to return to its lengthened state. This is called a restoring force: a force opposing the change to restore the system to its relaxed state. For most springs, the relationship between restoring force and displacement from the relaxed state is linear. For example, if we double the stretching (or compression), the force doubles. This relationship is called hooke"s law, and any spring obeying it is called an ideal spring. The negative sign is strictly a theoretical consideration to make the force vector in the opposite direction of the displacement. In free body diagrams you only need to worry about the magnitude (because you draw the vector in the correct direction). The coefficient k is called the spring constant and represents the sp(cid:396)i(cid:374)g"s stiff(cid:374)ess.

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