Problem 79 - A Complete Resource Book in Physics for JEE Main

Question

A stone of mass \(1.5 \mathrm{~kg}\) is tied at the end of \(0.5 \mathrm{~m}\) long string and whirled in a vertical circular path at constant speed of \(2 \mathrm{~ms}^{-1}\). The maximum tension in the string is \(\left(g=10 \mathrm{~ms}^{-2}\right)\) (A) \(27 \mathrm{~N}\) (B) \(3 \mathrm{~N}\) (C) \(90 \mathrm{~N}\) (D) \(15 \mathrm{~N}\)

Step-by-Step Solution

Answer

The maximum tension in the string is \(27 \, N\), so the correct answer is (A).

Step 1: Identify given quantities

We are given the following quantities: 1. Mass of the stone, \(m = 1.5 \, kg\) 2. Length of the string, \(L = 0.5 \, m\) 3. Constant speed of the stone, \(v = 2 \, ms^{-1}\) 4. Acceleration due to gravity, \(g = 10 \, ms^{-2}\) #Step2: Determine centrifugal force#

Step 2: Determine centrifugal force

The stone is moving in a vertical circular path with constant speed. Thus, it experiences a centrifugal force acting radially outward. The centrifugal force can be calculated using the following formula: \(F_{c}=m\frac{v^2}{r}\) where, \(m\) is the mass of the stone, \(v\) is the constant speed of the stone, and \(r\) is the radius of the circular path, which is equal to the length of the string. #Step3: Calculate centrifugal force at the bottom#

Step 3: Calculate centrifugal force at the bottom

We will first calculate the centrifugal force at the bottom of the circular path, using the given values for mass, speed, and length of the string: \(F_{c} = (1.5 \, kg) * \dfrac{(2 \, ms^{-1})^2}{0.5 \, m}\) \(F_{c} = 1.5 \cdot 4 / 0.5 = 12 \, N\) #Step4: Determine gravitational force at the bottom of the path#

Step 4: Determine gravitational force at the bottom of the path

When the stone is at the bottom of the circular path, the gravitational force acts downward, in the same direction as the centrifugal force. The gravitational force is given by the formula: \(F_{g} = mg\) #Step5: Calculate gravitational force#

Step 5: Calculate gravitational force

Using the given values for mass and acceleration due to gravity, we can calculate the gravitational force on the stone: \(F_{g} = (1.5 \, kg)(10 \, ms^{-2})\) \(F_{g} = 15 \, N\) #Step6: Determine the maximum tension in the string at the bottom#

Step 6: Determine the maximum tension in the string at the bottom

The maximum tension in the string occurs at the bottom of the circular path, where the gravitational force and the centrifugal force act in the same direction. The tension in the string must balance these two forces: \(T_{max} = F_{c} + F_{g}\) #Step7: Calculate the maximum tension in the string#

Step 7: Calculate the maximum tension in the string

Now, we can plug in the values we calculated and find the maximum tension in the string: \(T_{max} = 12 \, N + 15 \, N\) \(T_{max} = 27 \, N\) The maximum tension in the string is 27 N, so the correct answer is (A).