Question

A weightless thread can bear tension upto 37N. A stone of mass 500g is tied to it and revolves in a circular path of radius 4m in a vertical plane. If $g=10m{{s}^{-2}}$, then the maximum angular velocity of the stone will be.

A. 4 rad/s.
B. 2 rad/s.
C. 6 rad/s.
D. None of these.

Answer 1

- Hint: When a body moves in a circular path, a centripetal force is required to keep the body along its path. As per the question, the string is tied to a stone and the stone is moving in a vertical circular path. We will form an equation using a free body diagram and solve the equation further.

Formula used:
$\begin{align}
  & F=\dfrac{m{{v}^{2}}}{r} \\
 & F=mr{{\omega }^{2}} \\
\end{align}$

Complete step-by-step solution
The mass of the stone [m] = 500 gram i.e 0.5 kg.
The radius of the circular path is 4m.
We are asked to calculate the maximum angular velocity.
The centripetal force is calculated as –
$\begin{align}
  & F=\dfrac{m{{v}^{2}}}{r} \\
 & F=mr{{\omega }^{2}} \\
\end{align}$
Here v is tangential velocity and $\omega $ is the maximum angular velocity regarding v.
Maximum tension in the string will be at the lowest point of the circular path, where the tension will support the weight as well as provide the centripetal force for circular motion.
 So, we have
 $\begin{align}
  & T=mg+ml{{\omega }^{2}} \\
 & =ml{{\omega }^{2}}=T-5 \\
\end{align}$
For, maximum angular speed, the tension must be 37N
 So, the maximum angular speed is
 $\sqrt{\dfrac{T-5}{ml}}=\sqrt{\dfrac{37-5}{0.5\times 4}}=4rad/s$
 ${{\omega }_{\max }}=4rad/s$
 Hence, the maximum angular velocity of the stone will be $4rad/s$.

Therefore the correct option is A.

Note: When a body is rotated in a vertical circle, there are three forces acting upon the mass at all times; gravity, tension and centripetal force.
Centripetal force tends to move the mass away from the centre whereas the tension tends to pull the mass towards the centre. The centripetal force and tension are acting in opposite directions at all times, so they balance each other out leaving out gravity that determines the tension depending on the position of the mass.
When the mass is at the lowest position, the centripetal force acts downward (away from centre) while the tension acts upward (towards the centre). But since the force of gravity is always acting downward, the tension is loaded with two downward forces.
When the mass is at the highest position, the centripetal force acts upward (away from centre) while the tension acts downward (towards the centre). But since the force of gravity is again acting downward, the Centripetal force is loaded with two downward forces.