The time taken for the simple pendulum to complete an oscillation is referred to as the time period. Its frequency is the number of oscillations completed in one second, the unit being ‘hertz’ (Hz).
The length of a pendulum is the distance between the point of suspension and the centre of the bob:
The mass of the pendulum is obviously the total mass of the components that is mass of the bob and the mass of the string. However, the string has a very small mass which can be neglected hence I will consider the mass of the pendulum as the mass of the bob.
The amplitude is the maximum displacement of the bob when it is in motion. There are two types of amplitude regarding pendulums:
-
LINEAR AMPLITUDE which is the maximum linear distance travelled by the bob from rest position.
- ANGULAR AMPLITUDE which is the maximum angle displacement by the bob from its rest position.
The diagram below exhibits this:
forces acting on the pendulum
At rest position, there are two forces acting on the bob: the tension of the string on the bob acting upwards and the weight of the bob acting downwards. At rest position, these two forces act equal and opposite to each other (Newton’s Third Law of Motion) and so the pendulum is at equilibrium i.e. there is no overall change or effect.
However, when disturbed there is a change:
As shown in the diagram, the forces are equal but not opposite. Therefore there is a resultant force acting on the pendulum, which causes it to oscillate. In order to understand how the resultant force is brought about, I will resolve the force exerted by the weight of the bob into its components.
A force acting in one direction can be resolved into two mutually perpendicular components. When resolving a force one component force’s direction is selected and the other force is perpendicular to it. The component force does not necessarily have to be at 45 degrees to the force to be resolved:
To calculate the component forces, we have to make use of trigonometry:
Only force F and θ are known. To calculate F1 and F2 we can make use of the sine and cosine ratios:
Now I will apply this to the force of the weight of the pendulum:
The force W1 and the force of the tension of the string act equal and opposite to each other, so these forces are balanced out and there is no resultant force. However, the force W1 is not balanced by any force. Instead it acts as a resultant force on the pendulum and therefore the pendulum moves towards the rest position. This force is called the ‘restoring force’ as it tries to restore the object to equilibrium.
However, when it does reach the rest position the bob will not stop. Instead it will continue to ‘swing’ to the other side. This is because of inertia – the reluctance of a body to stop moving while it is in motion. It will continue to move to the other side and the process repeats, causing the pendulum to oscillate.
energy changes taking place during oscillation
At the rest position R the bob is at its minimum height from the ground. When the pendulum oscillates the height varies at different points. In the diagram A and B are at maximum height considering the height from position R as a reference. Since gravitational potential energy also depends on height, the pendulum gains maximum gravitational potential energy at positions A and B.
The gravitational potential energy is the product of the mass of the bob by the gravitational field strength of earth by the height of center of the bob from the reference level i.e. its height from the center of the bob during rest position.
