SIMPLE HARMONIC MOTION

One of the most common uses of oscillations has been in time-keeping purposes. In many modern clocks quartz is used for this purpose. However traditional clocks have made use of the pendulum. In this next section you will investigate how the motion of a pendulum depends on its physical characteristics.

The key feature of the motion is the time taken for one complete oscillation or swing of the pendulum. i.e. when the pendulum is again travelling in the same direction as the initial motion. The time taken for one complete oscillation is called the period.

Pendulum experiments

Your intuitive ideas

To begin your investigation you will need to set up a simple pendulum as shown in the diagram. You will need to be able to

• vary the length of the string;
• vary the mass on the end of the string;
• record the time taken for a particular number of

Once you are familiar with the apparatus try to decide which of the factors listed at the beginning of the next page affect the period. Do this without using the apparatus, but giving the answers that you intuitively expect.

The mass is attached by a string to the support, to form a simple pendulum.

• The length of the string
• The mass of the object on the end of the
• The initial starting position of the

Now try simple experiments to verify or disprove your intuitive ideas, using a table to record your results.

You are now in a position to start analysing the data obtained, using some of the basic mathematical concepts in pure

mathematics.

Analysis of results

You will probably have observed already that as you shortened the string the period decreased. Now you can begin to investigate

further the relationship between the length of the string and the period.

• Plot a graph for your results, showing period against length of string.
• Describe as fully as possible how the period varies with the length of the length of the string.

You may know from your knowledge of pure mathematics how a straight line can be obtained from your results using a log-log plot.

• Plot a graph of log (period) against log (length of string), and draw a line of best fit.
• Find the equation of the line you obtain and hence find the relationship between the period and the length of the length of the string.

Pendulum theory

You will have observed from Section 8.1 that the period of the

motion of a simple pendulum is approximately proportional to the square root of the length of the string. In this section you are presented with a theoretical approach to the problem.

The path of the mass is clearly an arc of a circle and so the results from Chapter 7, Circular Motion, will be of use here. It is convenient to use the unit vectors, er and eq , directed outwards along the radius and along the tangent respectively. The acceleration, a, of an object in circular motion is now given by

Forces acting on the pendulum

As in all mechanics problems, the first step you must take is to identify the forces acting. In this case there is the tension in the string and the force of gravity. There will, of course, also be air resistance, but you should assume that this is negligible in this case. The forces acting and their resultant are summarised in the table below.

Now it is possible to apply Newton’s second law, using the expression for the acceleration of an object in circular motion.

Solving the equation

Verify that

describes some aspect of the motion of the pendulum.

The variable, A, is known as the amplitude of the oscillation.

In this case the value of A is equal to the greatest angle that the string makes with the vertical.

It is sometimes also useful to talk about the frequency of an oscillation. This is defined as the number of oscillations per second.

The constant a is called the angle of phase, or simply the phase, and its value depends on the way in which the

pendulum is set in motion. If it is released from rest the angle of phase will be zero, but if it is flicked in some way, the angle of phase will have a non-zero value.

Energy consideration

An alternative approach is to use energy consideration.

As the simple pendulum moves there is an interchange of kinetic and potential energy. At the extremities of the swing there is zero kinetic energy and the maximum potential energy. At the lowest point of the swing the bob has its maximum kinetic energy and its minimum potential energy.

Energy

where T is the total energy of the system. The value of T can be found by considering the way in which the pendulum is set into motion.

Finding the speed

Solving the equation for v gives

This allows you to calculate the speed at any position of the pendulum.

You can also find an expression for from the equation for v.

Finding 

Simplifying the equation

If you assume that the oscillations of the pendulum are small, then you can use an approximation for cos q.

Small  approximation

Integrating the equation

You can solve this equation by separating the variables to give

The LHS is simply a standard integral that you can find in your tables book and the RHS is the integral of a constant.

This result is identical to that obtained earlier.

Modelling oscillations

In this section you will investigate other quantities which change with time, can be modelled as oscillations and can be described by an equation in the form

There are many other quantities that involve motion that can be described using this equation.

Some examples are the heights of tides, the motion of the needle in a sewing machine and the motion of the pistons in a car’s engine. In some cases the motion fits exactly the form given above, but in others it is a good approximation.

Fitting the equation to data

One example that could be modelled as an oscillation using the equation is the range of a tide (i.e. the difference between high and low tides). The table shows this range for a three-week period.

The graph below shows range against day.

Tidal range (metres)

Springs and oscillations

In Section 6.5, Hooke’s Law was used as the model that is universally accepted for describing the relationship between the tension and extension of a spring. Hooke’s Law states that

where T is the tension, k the spring stiffness constant and e the extension of the spring. If the force is measured in Newtons and the extension in metres, then k will have units Nm -1.

Masses can be attached to the spring suspended from the stand

To begin your investigations of the oscillations of a mass/spring system you will need to set up the apparatus as shown in the diagram opposite.

The equipment you will need is

• a stand to support your springs
• a variety of masses
• 2 or 3 identical springs
• a ruler a stopwatch

Intuitive ideas

If you pull down the mass a little and then release it, it will oscillate, up and down. Once you are familiar with the apparatus, try to decide how the factors listed below affect the period. Do this without using the apparatus, giving the answers that you intuitively expect.

• The mass attached to the
• The stiffness of the
• The initial displacement of the

Also consider some simple experiments to verify or disprove your intuitive ideas.

Theoretical analysis

To begin your theoretical analysis you need to identify the resultant force.

Explain why the resultant force is (mg – T )i.

Use Hooke’s Law to express T in terms of k, l and x, where k is the stiffness of the spring, l the natural length of the spring and x the length of the spring.

Use Newton’s second law to obtain the equation

Simple harmonic motion

If the equation describing the motion of an object is of the form

then that type of motion is described as simple harmonic motion (SHM).

This equation is deduced from the differential equation

Both the simple pendulum and the mass/spring system are examples of SHM.