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the period of a conical pendulum is term of its length semiveryical angle and acceleration due to gravity​

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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 __________

A. 12πlcosθg−−−−−√B. 12πlsinθg−−−−−√C. 4πlcosθ4g−−−−−√D. 4πltanθg−−−−−−√

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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.

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:

Resolving tension vector along x-axis and y-axis:

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×rh−−−−−−√v=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−−−−−√