episode 304: simple pendulum this episode reinforces many of the fundamental ideas about shm. note a complication: a simple pendulum shows

Episode 304: Simple pendulum
This episode reinforces many of the fundamental ideas about SHM.
Note a complication: a simple pendulum shows SHM only for small
amplitude oscillations.
Summary
Student experiment: Measuring the restoring force. (20 minutes)
Student experiment: Testing the relationship T = 2 √ (l/g). (30
minutes)
Student activity: Using an applet of a pendulum. (30 minutes)
Discussion: Gravitational and inertial mass. (10 minutes)
Student questions: Calculations involving pendulums. (30 minutes)
Student experiment:
Measuring the restoring force
Measure the restoring force for a simple pendulum.
TAP 304-1: The simple pendulum

Student experiment:
Testing the relationship T = 2 √ (l/g)
Test the relationship T = 2 √ (l/g) for a simple pendulum. Students
could decide for themselves which measurements to make, which
quantities to vary, and how to process and interpret the results.
Encourage them to look for deviations from linear behaviour, arising
from large-amplitude oscillations.
Student activity:
Using an applet of a pendulum
Investigate a virtual pendulum; this allows you to vary g. You can
also force the pendulum, which is useful later when studying
resonance.
http://monet.physik.unibas.ch/~elmer/pendulum/index.html
NB the analysis of the data uses log-log plots, so this may not be
suitable for all students
TAP 304-2: Virtual pendulum
Discussion:
Gravitational and inertial mass
The fact that the period of a simple pendulum is independent of the
mass of the bob is an example of the Principle of Equivalence –
something still not understood today and being tested by very
sophisticated experiments involving astronomical measurements on the
one hand and how single atoms fall due to gravity on the other.
The basic puzzle is why the m in F = ma (where m is the inertial mass
which determines how an object responds to any unbalanced force) has
exactly the same magnitude as the m in mg (where the m is the
gravitational mass, the source of the gravitational force).
In deriving the equation for the period of a simple pendulum, we have
used both, and used the fact that numerically they cancel out.
Student questions:
Calculations involving pendulums
These questions reinforce basic ideas about SHM.
TAP 304-3: Pendulum
TAP 304- 1: The simple pendulum
Requirements
*
large mass (at least 2 kg)
*
very strong cord
*
high, secure attachment for top of cord (tall retort stand is not
suitable)
*
spring balance (0–50 or 0–100 N)
*
metre rule with provision for horizontal clamping
Experimental exploration
With due regard to safety, and as far as laboratory conditions allow,
hang a large known mass m (several kg if possible) from a secure
attachment on 2 or 3 metres of strong cord. Clamp a metre rule
horizontal and level with the mid-point of the mass when it is hanging
vertically.
Attach a spring balance to the mass so that the force needed to pull
it sideways can be measured (i.e. the force needed to balance the
horizontal component of the tension). The diagram below shows the
arrangement.

Safety
In equilibrium the mass should be as close to the floor as possible
and the cord and support should be able to withstand many times the
weight of the mass. A soft landing platform must be under the mass at
all times, and feet kept well clear.
Wear safety spectacles to protect the eyes from the whipping end of a
broken cord.
Two persons are needed to fix a high suspension point: one to hold the
ladder or steps and one to do the work.

Record the length l of the string. Take a sequence of readings (F, x)
of the sideways force F needed to pull the mass a horizontal distance
x from equilibrium. Work up to as large a value of x as you judge
consistent with safety (remember, the tension in the cord will become
much bigger than the weight of the mass).
Plot a graph of F against x and comment on its shape. For the linear
region measure the ‘stiffness’ k (graph gradient) in N m-1.
Use your judgement to decide the value of x above which you could
consider the system non-linear. What value of angle  does this
correspond to?
Remember, sin  = x/l
Theory
Either from the triangle of forces, or by resolution horizontally and
vertically, show that
F = mg tan 
From tan x/h, but provided x is not too big (small ), h (which is
of course variable) is approximately the same as l (which is
constant), so that tan  = x/l
Hence
F = mgx/l
Another way of arriving at this is to use the small angle
approximation that tan   sin  for small .
You can now see how good the approximation of linearity is – it is as
good as you want it to be.
For  = 100 the difference between tan  and sin  is 1.5% of sin 
For  = 200 the difference becomes 6.4%
For  = 300 the difference becomes 15%
You can see that the theoretical ‘stiffness’ k is mg/l (for the linear
region).
Calculate this value and compare it with the value from the graph.
An interesting property of a pendulum is that its period is also
independent of its mass. Look at the theoretical expression for k and
see if you can see why this should be.
Practical advice
Students measure the restoring force on the bob of a simple pendulum
when it is displaced by different amounts. They then use trigonometry
and the small angle approximation to derive an expression for the
period of small-amplitude oscillations and consider what is meant here
by ‘small’.

Safety
In equilibrium the mass should be as close to the floor as possible
and the cord and support should be able to withstand many times the
weight of the mass. A soft landing platform must be under the mass at
all times, and feet kept well clear.
Wear safety spectacles to protect the eyes from the whipping end of a
broken cord.
Two persons are needed to fix a high suspension point: one to hold the
ladder or steps and one to do the work.
External reference
This activity is taken from Salters Horners Advanced Physics, section
BLD, additional sheet 4
TAP 304- 2: Virtual pendulum
Use a computer simulation of a pendulum to explore factors affecting
its oscillation.
Go to a suitable website with a simple pendulum simulation. For
example:
http://monet.physik.unibas.ch/~elmer/pendulum/index.html
http://www.walter-fendt.de/ph11e/pendulum.htm
Use the applet to demonstrate how the period, T, of a pendulum is
affected by its length, l, and/or the gravitational field strength, g.
Varying the length
*
Set g = 9.8 N kg-1
*
Select a value of l.
*
Find T by timing a known number of oscillations.
*
Change l, keeping g constant, and find the new T.
*
Continue until you have at least five sets of results.
*
Enter your results into a spreadsheet.
*
Plot a graph of log T against log l.
*
Find the gradient of the graph.
*
Hence obtain a relationship that shows how T varies with l.
Your relationship should be of the form T  la where the exponent a is
found from your graph.
Varying the gravitational field
The applet allows you to vary the gravitational field you cannot do in
a real Earth-based laboratory!
*
Select a value of l.
*
Choose a new value of g.
*
Find T by timing a known number of oscillations.
*
Change g, keeping l constant, and find the new T
*
Continue until you have at least five sets of results.
*
Enter your results into a spreadsheet.
*
Plot a graph of log T against log g.
*
Find the gradient of the graph.
*
Hence obtain a relationship that shows how T varies with g
Your relationship should be of the form T  gb where the exponent a is
found from your graph
Pendulum oscillations
Combine your two expressions to give a single expression connecting T,
l and g in the form:
T = klagb
Use one of your sets of ‘measurements’ to determine the value of the
constant k.
Use your complete formula to predict the period of a pendulum of
length = 2.8 m in a gravitational field g = 9.8 N kg-1
Check your prediction using the ‘virtual pendulum’.
Practical advice
Students will need Internet access. No other apparatus is needed. Use
of a spreadsheet to record and graph results may also be appropriate.
TAP 304- 3: Pendulum
The pendulum is often used as an example of a simple harmonic
oscillator. Consider an ideal pendulum consisting of a point mass, m,
in a gravitational field, g, at the end of a light string of length l.
The pendulum is displaced to one side of its rest position such that
the string makes an angle to the vertical of .
1. Draw a force diagram for the pendulum in this position.
2. In what direction does the resultant force act?
3. Derive an expression for the resultant force in terms of m, g and .
4. How does the force vary with angular displacement?
5. Why is the pendulum a good example of simple harmonic motion?
Under what conditions the pendulum could not be used as a good example
of simple harmonic motion?
Practical advice
These are interesting but challenging questions, to stretch more able
students.
Answers and worked solutions
1. Force diagram must show weight (mg) and the tension in the string,
e.g.:

2. The resultant force must act tangential to the arc traced out by
the swinging mass and hence perpendicular to the string.
3. Restoring force = component of weight perpendicular to the string

4. Force is a function of the sine of angular displacement.
5. This only works for simple harmonic motion where  is small so sin
 is considered to be the same as . The pendulum is not a simple
harmonic oscillator when the difference between sin and is too
large.
9

  • URINALYSIS BRIEF SHEET URINALYSIS COORDINATOROBSERVER RESPONSIBILITIES ARE SET FORTH
  • ZAŁĄCZNIK DO ZARZĄDZENIA NR 232021 PREZESA ZARZĄDU WFOŚIGW W
  • REGISTRATION LINE 6 BASIC TRAINING COURSE ON LEGAL
  • ENTREVISTA LUZ MARINA RODRIGUEZ VIVES VEREDA BORRACHERAS PROCESO HISTÓRICO
  • PAGE 7 PERSPECTIVES ON WARDECLARATION OF INDEPENDENCE C6S1 (THE
  • HERSHINOW 7 STEPHANIE INSLEY HERSHINOW STEPHANIEINSLEYGMAILCOM  14436045330 76
  • CHARACTER TRAITS TRUSTWORTHY DEPENDABLE RELIABLE LOYAL UNRELIABLE CHANGEABLE VARIABLE
  • AMPARO DIRECTO EN REVISIÓN 22952009 AMPARO DIRECTO EN REVISIÓN
  • DEUTSCHE GESELLSCHAFT FÜR AUSWÄRTIGE POLITIK REDE VON HERRN HALLDÓR
  • ANESZTEZIOLÓGIA SZÖVŐDMÉNYADATOK TECHNIKAI KÓD MEGNEVEZÉS ÖSSZES 1 DISCONNECTIO 2
  • BUMERANGS QUE RETORNEN PER JORDI BADOSA ELS PRIMERS BUMERANGS
  • MATEMATYCZNE PODSTAWY INFORMATYKI 15 ROZDZIAŁ 1MATEMATYCZNE PODSTAWY INFORMATYKI
  • 32000R2870 UREDBA KOMISIJE (EZ) BR 28702000 OD 19 PROSINCA
  • SYLABUS PRZEDMIOTU NAZWA PRZEDMIOTU METODY I TECHNIKI ORGANIZATORSKIE FORMA
  • SOCIALINĖS RIZIKOS GRUPĖS ASMENŲ VARTOJANČIŲ NARKOTINES IR PSICHOTROPINES MEDŽIAGAS
  • IES SANTA CLARA SISTEMAS AMBIENTALES Y SOCIEDADES 1º BACHILLER
  • COMPETENCYBASED CURRICULUM EXEMPLAR PGMA TRAINING FOR WORK SCHOLARSHIP SUPERMAID
  • ADI VE SOYADI 08012010CUMA 20092010 EMEK İLKÖĞRETİM OKULU 5B
  • 10030382_10030427_suministro_e_instalacion_de_dos_tableros_de_control_para_motores
  • SOLUBILITY REVIEW A SOLUTION IS A MIXTURE OF TWO
  • APOSTROPHE DIRECTIONS CIRCLE THE WORD IN PARENTHESES WHICH CORRECTLY
  • EK DOÇ DR DAVID AUSTIN PIERCE’IN ÖZGEÇMIŞI ADI SOYADI
  • NOTAT 08092006 200600214 – 08092006 200600214 – KOMMENTARER
  • MODELOS TEÓRICOS E INVESTIGACIÓN EN EL ÁMBITO DE LA
  • ACADEMIC WRITING TIP 26 THE APOSTROPHE HERE ARE A
  • CONFERENCIA NACIONAL DE GOBERNADORES COMISIÓN DE DESARROLLO DIGITAL COORDINACIÓN
  • APPENDIX 2 WEEKLY WORKING HOURS RECORD NAME POSITION SCHOOLRI
  • THE APOSTROPHE USE AN APOSTROPHE IN POSSESSIVE FORMS
  • PROCESO 111IP2015 INTERPRETACIÓN PREJUDICIAL DE LOS ARTÍCULOS 136 LITERAL
  • ABSTRAK PENULIS TERTARIK UNTUK MELAKUKAN PENELITIAN TENTANG PELAKSANAAN DISIPLIN