Question

A string of length 1m is fixed at one end with a bob of mass 100g and the string makes \[\dfrac{2}{\pi }\]rev/s around a vertical axis through a fixed point. The angle of inclination of the string with the vertical is?
A-\[{{\tan }^{-1}}(\dfrac{5}{8})\]
B-\[{{\tan }^{-1}}(\dfrac{3}{5})\]
C-\[{{\cos }^{-1}}(\dfrac{8}{5})\]
D-\[{{\cos }^{-1}}(\dfrac{5}{8})\]

Answer 1

Hint: The bob is rotating around the axis we are given with the frequency, by multiplying by \[2\pi \]we get the angular velocity. In order to find the angle that it made, we need to resolve and find out all the forces which are acting at it. At some point we have to find the different forces apart from gravitational force.

Complete step by step answer:
Given frequency, v= \[\dfrac{2}{\pi }\]rev/s
Thus, angular velocity, \[\omega =2\pi \times \dfrac{2}{\pi }=4\]rad/s
Let us assume that the angle made be p.
Centripetal forces are equal to centrifugal force which is equal to \[\dfrac{m{{v}^{2}}}{r}=mr{{\omega }^{2}}=m{{\omega }^{2}}l\sin p\]
We have resolved the forces the balanced forces can be written as:
\[mg=T\cos p\]---(1)
$ T\sin p=m{{\omega }^{2}}l\sin p \\ $
$ \Rightarrow T=m{{\omega }^{2}}l \\ $
Putting the value of T in eq (1), we get,
$ mg=m{{\omega }^{2}}l\cos p \\
 \Rightarrow \cos p=\dfrac{g}{{{\omega }^{2}}l} \\ $
Putting the values to find the value of angle,
$\cos p=\dfrac{10}{{{4}^{2}}\times 1} \\ $
$\Rightarrow \cos p=\dfrac{10}{16} \\ $
$\therefore p={{\cos }^{-1}}(\dfrac{5}{8}) \\$

So, the correct answer is “Option D”.

Note:
Since, the body is in circular motion there will act on its centripetal force and because the circular motion is preserved there will be some other force which is balancing the centripetal force. We had resolved the tension and made the second angle which is equal to the first because they are alternate angles. Tension is a force and is measured in Newtons. Tension has its origin in relation to the force of gravity.