Question

A particle of mass m is suspended from the ceiling through a string of length L. The particle moves in a horizontal circle of radius r. The speed of the particle is
options:
1)rg/root over L^2-r^2
2)r root g/L^2-r^2 whole power1/4
3)r root g/L^2-r^2 whole power 1/2
4)mgL/L^2-r^2 whole power 1/2

Answer 1

see attachment.....correct option is option (2)

Answer 2

Answer:

Option (2) is correct

Explanation:

The mass of the particle is m

The length of the ceiling is L

The radius of the circle is r

The angle made by the string with the normal of the ceiling is $\theta$

$\tan \theta=$               (1)

By balancing equation along horizontal axis.

$T \sin \theta=\dfrac{mv^2}{r}$                      (2)

The balancing equation along vertical axis

$T\cos \theta=mg$                                      (3)

Dividing equation 2 and 3 we get,

$\dfrac{T \sin \theta}{T \cos \theta}=\dfrac{mv^2}{mgr}\\\tan \theta=\dfrac{v^2}{rg}$                       (4)

Equating equation 1 and 4 we get,

$\dfrac{r}{\sqrt{l^2-r^2}}=\dfrac{v^2}{rg}\\v^2=\dfrac{r^2g}{\sqrt{l^2-r^2}}\\v=\dfrac{r \sqrt{g}}{(l^2-r^2)^{\dfrac{1}{4}}}$

So option (2) is correct.