define conical pendulum and give its derivation
From the point of view of a stationary external observer there are two external forces acting on a conical pendulum bob.
- the tension in the string which is exerted along the line of the string and acts towards the point of suspension
- the downward action of gravity on the mass of the bob
Tension in the string can be resolved vertically and horizontally:
- horizontal - Tsinθ (acting towards the centre of the circle)
- vertical - Tcosθ
Since the system is balanced in a vertical direction, the vertical component of the tension in the string is equal and opposite to the weight of the bob.
In a horizontal direction the system is imbalanced. The horizontal component of the tension in the string gives the blob an acceleration towards the centre of the circle.
Since the T in both equations refers to the same tension, both can be solved for T and equated. The mass, m, will divide out:
Since the speed of the pendulum bob is constant, it can be expressed as distance (circumference of the circle) over the time, t, required for one revolution of the bob (t here represents the period of the conical pendulum:
This is a simplification of setting all expressions equal to each other, followed by some algebraic manipulation:
These steps refer back to the equations set next to the diagram. They are used to simplify the equation so the period of the pendulum can be expressed in terms of its length or the radius of the circular path of the bob.