Question

A mass of $2\;kg\;$ is whirled in a horizontal circle by means of a string at an initial speed of $5\;rpm$. Keeping the radius constant, the tension in the string is doubled. The new speed is nearly
A. $14rpm$
B. $10rpm$
C. $7rpm$
D. $20rpm$

Answer 1

From the question, we can understand that the object is moving in a circular motion. Since tension is a kind of force, the tension acting on this object is equal to the centripetal force. So we can find the new speed that is the velocity using the centripetal force.

Complete step by step solution:
Given that the mass is $2\;kg\;$. Also, it is given that the object is moving with an initial speed of $5\;rpm$. We know that when an object is moving in a circular motion the force is provided by the centripetal force. This centripetal force gives us the tension acting on the string. Therefore we write this mathematically as,
${T_1} = \dfrac{{m{v_1}^2}}{{{r^2}}}$………. (1)
Here${v_1}^2$is the velocity with which the object is moving initially by the means of the string.
And then we keep the radius constant. So the radius will be $r$ in both the equations. Now the tension becomes double therefore we can write the second equation as,
$2{T_1} = \dfrac{{m{v_2}^2}}{{{r^2}}}$…………. (2)
Dividing equation (1) by (2). Since$m$ and $\;r$ are constant for both the cases they will be canceled out. So the remaining terms will be,
$\dfrac{{{T_1}}}{{2{T_1}}} = \dfrac{{{v_1}^2}}{{{v_2}^2}}$
${v_2}^2 = \dfrac{{2{T_1}}}{{{T_1}}}{v_1}^2$
${v_2} = \sqrt 2 {v_1}$
Substituting${v_1} = 5\;rpm$in the above equation we get,
${v_2} = \sqrt 2 \times 5\;rpm$
${v_2} = 7rpm$
Therefore the new velocity will be $7rpm$. Hence the correct option is C.

Note:
Here the unit used rpm is abbreviated as revolutions per minute. It is just what it sounds like. The number of complete turns that an object makes in one minute. We can always convert this unit from rpm to radians for convenience. A radian is actually a measure of an angle but can be defined in terms of $\pi$ to make calculations much easier in math and science in particular.