Question

A string will break under a load of $5\,Kg.$ to the one end of such a $2\,m$ long string a mass of $1\,Kg$ is attached. The maximum rpm in the horizontal plane so that the string does not break is $(g = 10\,m{s^{ - 2}})$
A. $28.66$
B. $47.73$
C. $38.92$
D. $54.12$

Answer 1

In order to find rpm of the string (rpm stands for rotation per minute), we will first find the magnitude of centripetal force acting on the load which is moving in circular path in horizontal plane and in order to continue this motion without breaking of string tis force will be balanced by the tension acting on string due to gravity.

Formula used:
Centripetal force on a body is calculated as
$F = mr{\omega ^2}$
where, $m$ is the mass of the body, $r$ is the radius of the circular path and $\omega$ is the angular velocity.
Frequency of the body is related with angular velocity as $\omega = 2\pi \nu$.

Complete step by step answer:
According to the question we have given that string will break under a load of $5\,Kg.$
Tension on the string due to acceleration due to gravity is $T = 5 \times g$
$T = 50N$
Now, the mass of the load is $m = 1\,Kg$. Length or radius of the circular path will be $r = 2m$. And let $\nu$ be the frequency of load, using formula of centripetal force $F = mr{\omega ^2}$ and putting $\omega = 2\pi \nu$ we get $F = 4{\pi ^2}{\nu ^2}mr$.

In order to balance the system $F = T$ hence, on putting the values we will get,
$4{\pi ^2}{\nu ^2}mr = 50$
$\Rightarrow {\nu ^2} = \dfrac{{50}}{{79}}$
$\Rightarrow \nu = \sqrt {0.632}$
$\Rightarrow \nu = 0.79{s^{ - 1}}$
Now, in order to find in rotations per minute we will multiply this value by $1\min = 60\sec$
So,
$rpm = 0.79 \times 60$
$\therefore rpm = 47.4\,{\min ^{ - 1}}$
And this value of rpm is very close to $47.73$.

Hence, the correct option is B.

Note: It should be remembered that, the load will continue to move in circular motion as long as the centripetal force on load will be balanced by tension acting on string, otherwise the string will break and here, if we use the exact value of acceleration due to gravity $g = 9.8m{s^{ - 2}}$ we would get the exact value of rpm given in the option $47.73$ but as in question its mentioned of using $(g = 10m{s^{ - 2}})$ that’s why our calculated value doesn’t exactly matched with the value given in option.