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.

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