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प्रश्न
What is a conical pendulum? Obtain an expression for its time period
उत्तर
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:
- 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) - 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 - 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 - 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) - 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)` - If T is the period of revolution of the bob, then
ω = `(2pi)/T = sqrt(g/(Lcosθ)`
∴ Period, T = `2pisqrt((Lcosθ)/g)`
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