Expression for tension in the string of a conical pendulum:
i. Consider a bob of mass ‘m’ tied to one end of a string of length ‘\(l\)’ and other end fixed to rigid support (S).
ii. Let the bob be displaced from its mean position and whirled around a horizontal circle of radius ‘r’ with constant angular velocity ‘ω’.
iii. During the motion, string is inclined to the vertical at an angle θ as shown in the figure.
iv. In the displaced position P, there are two forces acting on the bob:
a. The weight mg acting vertically downwards and
b. The tension T acting upwards along the string.
v. The tension (T) acting in the string can be resolved into two components:
a. T cos θ acting vertically upwards
b. T sin θ acting horizontally towards centre of the circle
vi. Vertical component T cos θ balances the weight of the bob and horizontal component Tsinθ provides the necessary centripetal force.
\(\therefore\) T cos θ = mg ….(1)
T sin θ = \(\cfrac{m\text v^2}r\)....(2)
vii. Squaring and adding equations (1) and (2),
viii. Dividing equation (2) by (1),
ix. From equation (3) and (4),
This is required expression for tension in the string.
