
The period of a conical pendulum in terms of its length , semi vertical angle and acceleration due to gravity is __________
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.
is the distance from the plane of the circular motion to where the string is attached
is the radius of the circular path in meters
is the angle between h and the string in degrees
is the length of the string in meters
is the mass of mass at the end of the string in kilograms
is the angular velocity of the mass in radians-per-second
Using Newton’s second law of motion,
Where,
is the force acting on a body
is the mass of the body
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 , where the acceleration of the mass will be zero because the mass is not accelerating vertically.
Finding the tension in the string,
And,
Eliminating , we get,
We get,
We have,
Therefore,
We have,
Therefore,
Or,
Time period of conical pendulum is given as,
Or,
The period of a conical pendulum is given as
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.
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.

Using Newton’s second law of motion,
Where,
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
Finding the tension in the string,
And,
Eliminating
We get,
We have,
Therefore,
We have,
Therefore,
Or,
Time period of conical pendulum is given as,
Or,
The period of a conical pendulum is given as
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.
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