Simple Harmonic Motion

Some of the motions we encounter are of repetitive nature, these motions are called oscillations. Some examples of these motions are : the swing of a pendulum, the vibrations of a guitar string or the diaphragms of speaker systems. Some other forms of oscillations which are less obvious or evident are those of air molecules transmitting sounds, the atoms of solids conveying temperature or the electrons in antennas. We are now going to study some of the properties of such periodic oscillations, often called Harmonic Motions.

The simple pendulum consists of an idealized body-a point mass suspended from a massless inextensible string swinging in a vertical plane solely under the influence of gravity. In this experiment we are going to use the small angle approximation

Tetta=Sin(Tetta)

which is true for small values of Tetta for Tetta in radians. In this case the restoring force exerted on the simple pendulum. A body will oscillate in linear simple harmonic motion if it is acted upon by a restoring force whose magnitude is proportional to the linear displacement of the body from its equilibrium position (F = -kx). Similarly, a body will oscillate in rotational simple harmonic motion if there is a restoring torque that is proportional to the angular displacement of the body from its equilibrium position. For this experiment, you will explore both kinds of harmonic motion.

You will use the most common example of linear simple harmonic motion, an object attached to a spring.

The most common example of rotational simple harmonic motion is the simple pendulum where, for small angular displacements of the pendulum from the equilibrium (vertical) position, the restoring torque is approximately proportional to the angle of displacement.

We have chosen another example for this experiment, the torsion pendulum. A long cylindrical metal rod is held fixed in a support bracket with a heavy cylindrical plate attached to its lower end . If you rotate the plate through an angle è, the rod will exert on the plate a restoring torque that is proportional to Tetta, as long as è does not exceed the elastic limit of the rod.

If you rotate the plate and release it, the plate will oscillate in simple harmonic motion. From measurements of the period of the oscillation and the geometry of the rod and plate, you can calculate the torsion modulus, a quantity that characterizes the resistance of a material to twisting. Torsion pendulum (rotational) oscillator.

Its importance is that it can serve as a mathematical model of a variety of systems and provides the basis of the characterisation of more complicated motions through the techniques of Fourier analysis. Simple harmonic motion can in some cases be considered to be the one-dimensional projection of two-dimensional circular motion.

Consider a long pendulum swinging over the turntable of a record player. On the edge of the turntable there is an object. If the object is viewed from the same level as the turntable, a projection of the motion of the object seems to be moving backwards and forwards on a straight line. It is possible to change the frequency of rotation of the turntable in order to have a perfect synchronization with the motion of the pendulum.