Question

5) For a conical pendulum prove that tantheta=vsquare /rg​

Answer 1

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Answer 2

Answer:

The mathematical proof of the statement that tan(theta) = v^2/rg for a conical pendulum.

Explanation:

A conical pendulum is a pendulum in which the bob is attached to the end of a string of length L that is free to move in a vertical plane. The string is wrapped around a fixed point O on the rim of a vertical cone, and the bob hangs vertically beneath the point of suspension P. The angle theta between the vertical and the string is defined as the angle between the vertical and the line segment OP.

To prove that tan(theta) = v^2/rg, we can use the conservation of energy principle, which states that the total mechanical energy of the system (the sum of kinetic and potential energy) remains constant as long as there are no external forces acting on the system.

At the bottom of the swing (theta = 90 degrees), the bob has maximum potential energy and no kinetic energy. As the bob swings up, some of its potential energy is converted to kinetic energy, and at the top of the swing (theta = 0 degrees), the bob has maximum kinetic energy and no potential energy.

At any point in the swing, the potential energy is given by mgh, where m is the mass of the bob, g is the acceleration due to gravity, and h is the height of the bob above the point O. The kinetic energy is given by (1/2)mv^2, where v is the velocity of the bob.

As the bob swings, the height h is given by the expression h = Lcos(theta), and the velocity v is given by the expression v = (L/T)sin(theta), where T is the period of the pendulum.

Substituting these expressions into the equations for kinetic and potential energy, we get:

Potential energy = mglcos(theta)

Kinetic energy = (1/2)m((L/T)sin(theta))^2

And by conservation of energy, we have:

mglcos(theta) = (1/2)m((L/T)sin(theta))^2

and rearranging we get

tan(theta) = (L/T)^2*sin(theta)cos(theta)/g

and as sin(theta) = v/L and cos(theta) = r/L, we can substitute these values to get

tan(theta) = v^2/rg

This is the mathematical proof of the statement that tan(theta) = v^2/rg for a conical pendulum.

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