Bartons Pendulum

Oscilations

Pendula

A pendulum consists of an arm of low mass with a bob, which has a higher mass, on the end. The top of the arm is pivoted so that the pendulum can swing. A pendulum will continue to swing back and forth indefinately, until it is stopped by air friction and friction within the pivot. When the angle, Ñ"б, between the stationary line of the pendulum (the line towards which the movement tends) and the line of maximum amplitude of the pendulum is quite small, then the time period of the pendulum can be found according to the following equation:

, where l is the length of the arm of the pendulum (between the pivot and the centre of mass of the bob) and g is the acceleration due to gravity (on earth Ñ"с„lÑ"n9.81). For the spring, a similar equation can be derived. For any spring, , where m is the mass of the bob on the spring and k is hookes constant. Hookes constant is the constant of proportionality of force against extension

for any spring, and varies from spring to spring. In formulaic terms. . The unit for thisd quantity is newtons per metre. Substituting the above equation (Hooke\\\\\\\'s Law) into the equation, , and therefore, . g here is the acceleration due to gravity, as the force on the spring consists of the weight of the bob. On the moon, the time period of the pendulum would change, as l is a constant where as g would change, where as the time period of the spring would stay constant, as is a constant, and x changes proportionally to g.

What connects the motion of both the spring and the pendulum? They are connected by the fact that they both move with simple harmonic motion (SHM), which is the most common form of motion, as it is related to circular motion, and also to the motion of pendula, springs, logs in th water, water in a u-tube, pulsars, vibrating molecules, etc.

What is simple harmonic motion? it is motion in which:

1. The motion (and the acceleration) is directed towards a fixed point.

2. The acceleration is proportional to the displacement.

This leads to the general equation for simple harmonic motion - . In this equation Ñ"Ð* is the angular velocity. The - sign shows that the acceleration is in the opposite direction to the displacement.

Consider a pendulum. The further away the bob is from the centre of the swing, the faster it is slowing down (at the point when the pendulum hangs vertically, there is no acceleration on the pendulum at all.) This acceleration is always acting in the opposite direction to the displacement as well - it is pulling the pendulum in towards a central point. And the acceleration is directed towards a fixed point- it takes place towards the fidicial point and the line which is drawn vertically down from the pivot.

It is a useful exercise to derive the equation for the time period of a pendulum given above, as follows.

At any point in the swing of a pendulum, there are two forces acting on the pendulum - the force of its weight and the tension in the string connecting it to the pivot. The resultant of these two forces is the force acting on the bob pf the pendulum. As we know that this is simple harmonioc motion, then the force must be in the direction of the fidicial point.

This force must be equal to the weight x sin Ñ"б. As this is an equilateral triangle with sides l and x, x=lÑ"б, and therefore . Therefore, we can say that , as the acceleration takes place in the opposite direction to the displacement. Simplifying, .

As this is simple harmonic motion, the equation applies to the equation, which can be substituted into the original formula, as follows: . Substituting the general time period equation for circular motion, , giving us the required formula

Graphs of Simple Harmonic Motion.

Displacement, time

These graphs are taken by considering a mass on the end of a string being whirled in a vertical circle, and an eye prescisely in the plane of the motion observing it. This gives a graph of the vertial displacementof the object:

This is a sine curve, so , where a is the maximum amplitude or the radius of the circle and Ñ"б is the angle. From the definition of angular velocity we get the equation , and therefore and ., This is the displacement equation for simple harmonic motion.

Example

A pendulum with a 2m long arm is oscilating with a time period aof 2 seconds. Find the displacement of the bob from the fidicial point at t=1.2s.

.

2. Velocity, time

The velocity is in the vertical direction as the displacement was measured in the vertical direction. As , and , you would expect the graph to be a graph of , and it is. v can also be expressed in the formula

3. Acceleration, time

Again, the acceleration is measured in a vertical direction. The graph is a graph if , as

The parameters change as the pendulum swings. When the pendulum is in its equilibrium position, x and a are both 0. However, v is at a maximum, Ñ"Ð*a. As the pendulum moves off to the right, x increases. However, as x increases, the acceleration increases in magnitude as well (because the pendulumÐŽ¦s wieght is not matched by the tension in the string). For this reason, the acceleration becomes negative, as it is always in the opposite direction to the displacement, and therefore the velocity goes down. At the maximum height of the swing to the right, x=a, the amplitude of the swing, v=0 and a=-Ñ"Ð*2r. Therefore, the pendulum then starts to swing back to the left, the velocity decreasing all the time (under 0), x decreasing towards 0, and a decreasing until, when the pendulum is back in equilibrium position, x=0, a=0, and v=-Ñ"Ð*a. As the pendulum swings to the left, the same

things happen, except that the signs are the other way round, the v and x being negative and a being positive.

Damping

Unless there is a source of energy constantly maintaining a vibration, the amplitude of the vibration becomes smaller and smaller. This is called damped motion. This is because some of the energy in the system is used to overcome resistive forces - e.g. air friction in a simple pendulum. This is natural