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, á, 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 ñln9.81). 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 ç 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 á. As this is an equilateral triangle with sides l and x, x=lá, 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.
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...