A pendulum is a system composed of a mass coupled to a pivot, which allows its movement freely. The mass is subject to the restorative force caused by gravity.
There are numerous pendulums studied by physicists, as they describe it as an object of easy prediction of movement and which has enabled numerous technological advances. Some of these are the physical, twisting, tapered, Foucalt, double, spiral, Karter and inverted pendulums. But the simplest and most widely used model is the simple pendulum.
This pendulum consists of a mass attached to a flexible and inextensible thread at one end and free at the other, represented as follows:
When we move the mass out of its resting position and release it, the pendulum makes oscillations. When we disregard air resistance, the only forces acting on the pendulum are the tension with the wire and the weight of the mass. m. Thus:
The Weight force component that is given by P.cosθ will nullify with the Wire Tension force, so the only cause of oscillatory motion is the P.senθ. So:
However, the angle θ, expressed in radians which by definition is given by the arc quotient described by the angle, which in the oscillatory motion of a pendulum is x and the radius of its application, in this case, given by ℓ, like this:
Where to replace in F:
Thus it can be concluded that the movement of a simple pendulum does not describe an MHS, since the force is not proportional to the elongation but to its sine. However, for small angles, , the sine value of the angle is approximately equal to this angle.
So when we consider the cases of small oscillation angles:
As P = mg, and m, g and ℓ are constant in this system we can consider that:
So we rewrote the restorative force of the system as:
Thus, the analysis of a simple pendulum shows us that for small oscillations a simple pendulum describes an MHS.
As for any MHS, the period is given by:
It is like
Then the period of a simple pendulum can be expressed by: