Question

a conical pendulum has period of one second and its string makes an angle of 45 degree with the vertical.if the same pendulum is used as simple pendulum what will be its period?

Answer 1

Answer on Question #57702, Physics / Mechanics | Relativity

A conical pendulum has period of one second and its string makes an angle of 45 degree with the vertical. If the same pendulum is used as simple pendulum what will be its period?

Solution:

The period of revolution of conical pendulum on a string of length $L$ at an angle of $\theta$ from the vertical is

$$T_1 = 2\pi \sqrt{\frac{L \cos \theta}{g}}$$

For small amplitudes, the period of simple pendulum can be approximated by:

$$T_2 = 2\pi \sqrt{\frac{L}{g}}$$

From first equation, we have

$$T_1 = 2\pi \sqrt{\frac{L}{g}} \cdot \sqrt{\cos \theta} = T_2 \sqrt{\cos \theta}$$

Hence,

$$T_2 = \frac{T_1}{\sqrt{\cos \theta}} = \frac{1}{\sqrt{\cos 45{}^\circ}} = 1.189\, s \approx 1.2\, s$$

Answer: $1.189\, s$

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