Oscillating Systems

Oscillations & Waves, Reflection & Refraction

OSCILLATING SYSTEMS (return to CONTENTS)

TERMINOLOGY (return to start of page)

An oscillation (or vibration) involves a particular cycle of motion being repeated over and over.

O is the rest or equilibrium position
The amplitude is the size of the maximum displacement = OA or OB (not A to B)
One cycle is one complete oscillation: E.g. (O to A to B to O) or (A to B to A)
The period (T) is the time for one cycle (usually in seconds)
The frequency (f) is the number of cycles per second (‘cycles per second’ is called hertz (Hz))

Example

If 5 cycles of a vibration take one second, then one cycle takes 1/5 = 0.2 seconds. In general:

Measuring period

We could determine the period of a pendulum or mass on a spring by timing a single cycle. However, timing just one cycle would be inaccurate. Rather, we would time, say, 20 cycles and divide by 20.

E.g.. For a particular pendulum, 20 cycles takes 15 sec \ 20T = 15 s, \ period, T = 15/20 = 0.75 sec.

SIMPLE HARMONIC MOTION (‘SHM’) (return to start of page)

In the diagram below, point N is the projection of point P onto the line JK. Line PN is always at right angles (90⁰) to JK.
P moves uniformly round the circle of radius A. As P goes round and round, the point N moves up and down the line JK.
Point N is repeating the same basic motion, or cycle, over and over. Thus, we have a close link between the circular motion of point P and oscillations along a straight line of point N.

The amplitude of the motion of N equals the radius of the circle, A.

Motion of N

Displacement, x

Velocity, v

It can be shown, from the expression for x, that v is related to time by:

And v is related to position, i.e. distance x from the centre, by:

Acceleration and the conditions for SHM

We can also show, from the expression for v, that the acceleration a is related to time t by:

An object is said to move with simple harmonic motion if its acceleration is given by an equation of the above form.

The above equation indicates that:

when x is positive (N is above the centre) a is negative
when x is negative (N is below the centre) a is positive

Thus, the acceleration is always directed towards a fixed point (this being the centre, where x = 0) , and is directly proportional to the distance from the centre.

Thus, the conditions for simple harmonic motion can be expressed as:

A body undergoes simple harmonic motion if its acceleration is always directed towards a fixed point and is directly proportional to its distance from that point

A body which moves with simple harmonic motion is called a simple harmonic oscillator.

Graphical representations

Variations with time

Variations with distance

As already stated:

The following represents graphs for a-x and v-x superimposed:

Example

A coin lies on a surface which is vibrating vertically with SHM of amplitude 5cm. At what frequency will the coin begin to lose contact with the surface, and start to 'chatter'. (g = 10 ms^-2)

Amplitude = 5cm = 0.05m

The coin will lose contact with the surface if the surface accelerates downwards at a greater rate than the coin, i.e. if:

MASS ON A SPRING (return to start of page)

l = the extension produced when the mass m is attached to the spring. If the mass is pulled a little further down and released, it will oscillate about its equilibrium position. At a distance x from equilibrium, the mass experiences a ‘restoring force’ F, i.e. one trying to force it back to the equilibrium position.

The constant k is called the ‘spring constant’ of the spring (Þ k = F/x, unit Nm^-1)

We see from F = kx that the bigger the value of k the bigger the force F needed to get a given extension x. Thus, k is a measure of the stiffness of the spring, i.e. a measure of how hard it is to stretch (or compress) it.

The expression F = kx gives the magnitude of the restoring force.

when the mass is below the equilibrium position F points upwards
when the mass is above the equilibrium position F points downwards

So, wherever the mass is, the force always acts towards the same point, the equilibrium point.

We can take the relative directions of F and x into account by writing:

F and a have the same direction as each other, and are always opposite to x:

Now, in the above equation for acceleration a, (k/m) is a positive constant for a given spring and mass, so the equation is of the form:

Thus, a mass oscillating on a spring satisfies the conditions for SHM. The mass oscillates just like the point N in the earlier analysis.

The power ‘½’ representing ‘square-root’ of the content of the brackets.

In the position shown in (b) for the mass-spring system above, the mass is at its equilibrium position. In this position:

Where, l = original extension produced when the mass m is hung on the spring.

SIMPLE PENDULUM (return to start of page)

This consists of a small bob on a thread of a fixed length.

* For example, find sin10⁰and then convert 10⁰to radians.

To indicate the relative directions of F and x, we write:

Thus, the pendulum bob moves with simple harmonic motion.

This equation is identical to that for a mass on a spring, but notice that the meaning of l is different in each case.

Experiment

A simple ‘at-home’ experiment can be performed to use the above to measure g.

Make a simple pendulum, say, 1 m long
Time, say 20, oscillations (using small amplitude oscillations)
Hence, calculate the period T
Hence, determine a value for g using the above equation (the accepted value is 9.8ms^-2)

ENERGY IN SHM (return to start of page)

Consider a mass m oscillating between J and K with SHM of amplitude A. Point O is the equilibrium position.

Kinetic energy (KE)

At a displacement x:

Total energy

In a free oscillation (i.e. one which neither gains nor loses energy) the total energy (= kinetic + potential) is constant, i.e. the same at every point in the oscillation. So if we know it at one point, we know it at all points.

In the above oscillation, at the middle, PE = 0 and KE = its maximum, \ KE at middle = total energy
Thus, total energy = KE when x = 0, so from the above equation:

Potential energy

Notice that, while KE and PE both depend upon x, the total energy does not, since its value is the same at every point, i.e. for any x value.

Graphical representation

Example

Show that PE = KE when x = A/Ö2, as indicated above.

Energy in a spring

In both an oscillating pendulum and a mass-spring system there is a continuous exchange between PE and KE. If the oscillations are 'free', then the sum of these is constant, and is given by:

For a mass-spring system, we can express this in terms of the spring constant k.

This equals the energy when the spring is stretched to its maximum, i.e. when the extension equals the amplitude, A.

Now, the strain energy stored in a spring when it is stretched by an amount A by a force F is given by:

Damping

A free vibration is one which continues with constant amplitude, neither gaining nor losing energy.
For such an oscillation a displacement (x) against time (t) curve might look like:

An oscillation that loses its energy is said to be damped.

a) Light damping

Ping-Pong ball pendulum - the air provides light damping

b) Heavy damping

c) Critical damping

This is said to occur when the time for the displacement to become zero is a minimum:

Damping is made use of in a number of devices. In a car, for example, the suspension makes the ride smoother when passing over bumps, but it has to be damped to prevent the car bouncing too much after a bump has been passed.

FORCED VIBRATIONS AND RESONANCE (return to start of page)

A system that can vibrate, such as a pendulum or a mass on a spring, has a natural period, this being the period with which it vibrates if it is set in motion and then left to oscillate. It also has a corresponding natural frequency. Some systems have a number of natural periods/frequencies.

When you push a child's swing you automatically push it, i.e. apply a forcing vibration, in time with its own natural period, and this allows the amplitude of the swing to build up. This is an example of resonance.

Resonance is said to occur when a system is forced to vibrate at its own natural frequency, and this can produce large amplitude vibrations

Observation

Pendulum A is set in motion.

We see that:

The vibration carries along the top string (which should be a little loose). All the other three pendulums are affected – but pendulum C picks up the biggest amplitude vibration.

We infer that:

Pendulum A applies a forcing frequency to all the other three.

However, pendulum A and C are the same length, and so have the same natural period and the same natural frequency. So resonance occurs between A and C, producing a large vibration of C.

B and D have different lengths and therefore different natural frequencies to A and so resonance does not occur between these and A.

Graphical representation

Maximum amplitude occurs when:

Note: Resonance is a potentially dangerous phenomenon. In 1940 the Tacoma Narrows Suspension Bridge collapsed, just a few months after its opening. When subject to a quite moderate wind it was set in vibratory motion, and the amplitude of vibration built up till the bridge collapsed. Engineers have to be aware of possible resonance occurring when structures are designed.

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