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The period of a conical pendulum in terms of its length (l), semi vertical angle (θ) and acceleration due to gravity (g) is __________
A12πlcosθgB12πlsinθgC. 4πlcosθ4gD. 4πltanθg

Answer
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Hint: A conical pendulum has a mass attached to a string along the vertical. The mass executes circular motion in the horizontal plane. For calculating the time period of a conical pendulum, we need to use the expression of Newton's second law of motion. The tension along the string can be resolved in the x-axis and the y-axis. Along vertical, there should be no acceleration of the mass.

Complete answer:
A conical pendulum is a system of a mass attached to a nearly massless string that is held at the opposite end and swung in the horizontal circles. It doesn't take much effort to keep the mass moving at a constant angular velocity in a circle of constant radius.

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h is the distance from the plane of the circular motion to where the string is attached
r is the radius of the circular path in meters
θ is the angle between h and the string in degrees
l is the length of the string in meters
m is the mass of mass at the end of the string in kilograms
ω is the angular velocity of the mass in radians-per-second
Using Newton’s second law of motion,
F=ma
Where,
F is the force acting on a body
m is the mass of the body
a is the acceleration of the body

Force-body diagram of the mass of the pendulum:
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Resolving tension vector along x-axis and y-axis:
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Let's plug in the vertical forces acting on the mass into F=ma , where the acceleration of the mass will be zero because the mass is not accelerating vertically.
Finding the tension in the string,
Tcosθ=mg
And,
Tsinθ=mv2r
Eliminating T, we get,
Tanθ=v2rg
We get,
v=rgtanθ
We have,
tanθ=rh
Therefore,
v=rg×rhv=r2g×lhl
We have,
cosθ=hl
Therefore,
v=r2glcosθ
Or,
vr=glcosθ
Time period of conical pendulum is given as,
T=2πrvT=2πlcosθg
Or,
T=4πlcosθ4g
The period of a conical pendulum is given as T=4πlcosθ4g

So, the correct answer is “Option C”.

Note:
A simple pendulum is a special case of a conical pendulum in which the angle made by the string with the vertical is zero. Time period of a pendulum can be calculated by applying Newton's second law of motion, which gives us the time period of a pendulum in terms of length of string, semi vertical angle and the radius of the circular path.