Question

A body of mass 4 kg is tied to one end of a rope of length 40 cm and whirled in a horizontal circle. The maximum number of revolutions per minute it can be whirled so that the rope does not snap as the rope can withstand to a tension of 6.4 Newton, will be –

A. 1.91
B. 19.1
C. 191
D. 1910

Answer 1

Hint: Firstly, we will find the value of the velocity of the body using the formula that relates the tension in the rope with the length of the rope, the mass and velocity of the body. Using this velocity of the body, we will find the number of revolutions per minute by using the formula that relates the angular frequency with the velocity of the body and length of the rope.

Formula used:
\[\begin{align}
  & T=\dfrac{m{{v}^{2}}}{r} \\
 & w=\dfrac{v}{r} \\
\end{align}\]

Complete step by step answer:
From the data, we have the data as follows.
The mass of the body, m = 4 kg
The length of the string = the radius of the circle it forms, r = 40 cm
r = 0.4 m
The tension in the rope, T = 6.4 N
The formula used to calculate the tension in the rope is,
\[T=\dfrac{m{{v}^{2}}}{r}\]
Where m is the mass of the body, v is the velocity of the body and r is the length of the rope.

A diagram representing the tension in the rope is,

Substitute the given values in the above equation.
\[\begin{align}
  & 6.4=\dfrac{4\times {{v}^{2}}}{0.4} \\
 & \Rightarrow {{v}^{2}}=\dfrac{6.4\times 0.4}{4} \\
\end{align}\]

Continue further calculation.
\[\begin{align}
  & {{v}^{2}}=0.64 \\
 & \Rightarrow v=0.8\dfrac{m}{s} \\
\end{align}\]
Now, consider the formula that relates the angular frequency with the velocity of the body and the length of the rope.
\[w=\dfrac{v}{r}\]
Where v is the velocity of the body and r is the length of the rope.
Substitute the given values in the above equation.
\[\begin{align}
  & w=\dfrac{0.8}{0.4} \\
 & \Rightarrow w=0.2\dfrac{rad}{\sec } \\
\end{align}\]
Continue the further calculation to convert the unit from rad per sec to rev per min. So, we have,
\[\begin{align}
  & w=\dfrac{0.2}{2\pi }\times 60 \\
 & \Rightarrow w=1.91\dfrac{rev}{\min } \\
\end{align}\]
The maximum number of revolutions per minute is 1.91.
As the maximum number of revolutions per minute it can be whirled so that the rope does not snap as the rope can withstand a tension of 6.4 Newton, will be 1.91, thus, the option (A) is correct.

Note:
The units of the parameters should be taken care of. In this case, we have converted the unit of the radius of the circle, that is, the length of tope from cm to m. And even, we have converted the unit of the angular frequency from rad per sec to rev per min, so, this one should be known.