Question

What is a conical pendulum? Obtain an expression for its time period

Answer 1

A tiny mass (assumed to be a point object and called a bob) connected to a long, flexible, massless, inextensible string, and suspended to rigid support revolves in such a way that the string moves along the surface of a right circular cone of the vertical axis and the point object performs a uniform horizontal circular motion. Such a system is called a conical pendulum. 

Expression for its time period:

1. Consider the vertical section of a conical pendulum having bob (point mass) of mass m and string of length ‘L’.
   Here, θ is the angle made by the string with the vertical, at any position (semi-vertical angle of the cone)
2. In a given position B, the forces acting on the bob are
   a. its weight ‘mg’ directed vertically downwards
   b. the force ‘T₀’ due to the tension in the string, directed along the string, towards the support A.
   
   In an inertial frame
3. As the motion of the bob is a horizontal circular motion, the resultant force must be horizontal and directed towards the centre C of the circular motion.
   For this, tension (T₀) in the string is resolved into
   a. T₀ cos θ: vertical component
   b. T₀ sin θ: horizontal component
4. The vertical component (T₀ cos θ) balances the weight ‘mg’.
   ∴ mg = T₀ cosθ …..............(1)
   The horizontal component T₀ sinθ then becomes the resultant force which is centripetal.
   mrω² = T₀ sinθ …..............(2)
   Dividing equation (2) by equation (1),
   ω² = `(gsinθ)/(rcosθ)` …..............(3)
5. From the figure,
   sinθ = `r/L`
   ∴ r = L sin θ …..............(4)
   From equation (3) and (4),
   ω² = `(gsinθ)/(L.sinθ.cosθ)`
   ω = `sqrt(g/(Lcosθ)`
6. If T is the period of revolution of the bob, then
   ω = `(2pi)/T = sqrt(g/(Lcosθ)`
   ∴ Period, T = `2pisqrt((Lcosθ)/g)`