Problem 82 - A Complete Resource Book in Physics for JEE Main

Question

A \(2 \mathrm{~kg}\) stone at the end of a string \(1 \mathrm{~m}\) long is whirled in a vertical circle at a constant speed. The speed of the stone is \(4 \mathrm{~m} / \mathrm{s}\). The tension in the string will be \(52 \mathrm{~N}\), when the stone is (A) at the top of the circle (B) at the bottom of the circle (C) halfway down (D) none of the above

Step-by-Step Solution

Answer

The tension in the string will be \(52\,\mathrm{N}\) when the stone is at none of the given positions, as shown by the calculations: \(T_t = 51.6\,\mathrm{N}\) at the top, \(T_b = 12.4\,\mathrm{N}\) at the bottom, and a complex calculation for the halfway position. Therefore, the correct answer is (D) none of the above.

Step 1: Identify the forces acting on the stone

There are two forces acting on the stone: 1. Gravitational force: \(F_g = mg\), where m is the mass of the stone (2 kg) and g is the acceleration due to gravity (9.8 m/s^2). 2. Tension force: \(F_t = T\), where T is the tension in the string (52 N).

Step 2: Calculate the gravitational force acting on the stone

Using the formula for gravitational force, we can find the force acting on the stone: \(F_g = mg = (2\,\text{kg})(9.8\,\text{m/s}^2) = 19.6\,\mathrm{N}\)

Step 3: Calculate the centripetal force acting on the stone

We can calculate the stone's centripetal force with the formula \(F_c = \dfrac{mv^2}{r}\), where \(m\) is the mass (2 kg), \(v\) is the constant speed of the stone (4 m/s), and \(r\) is the length of the string (1 m). Using the values, we get: \(F_c = \dfrac{mv^2}{r} = \dfrac{(2\,\text{kg})(4\,\text{m/s})^2}{1\,\text{m}} = \dfrac{32\,\text{N}}{1\,\text{m}} = 32\,\mathrm{N}\)

Step 4: Analyze net forces acting on the stone at different positions of the circle

1. At the top of the circle: The gravitational force and centripetal force will act in the same direction (downward), so the tension will have to balance both forces: \(T_t = F_c + F_g\) 2. At the bottom of the circle: The gravitational force and centripetal force will act in opposite directions. Therefore, the tension will have to balance the difference between the forces: \(T_b = F_c - F_g\) 3. Halfway down the circle: In this case, net force calculations become more complex because they're no longer in the same line.

Step 5: Find the position of the stone that corresponds to a tension of 52 N

1. At the top of the circle: \(T_t = F_c + F_g = 32\,\mathrm{N} + 19.6\,\mathrm{N} = 51.6\,\mathrm{N}\) 2. At the bottom of the circle: \(T_b = F_c - F_g = 32\,\mathrm{N} - 19.6\,\mathrm{N} = 12.4\,\mathrm{N}\) Neither of these tensions are equal to the given tension (52 N). Therefore, the stone must be at a position other than the top or the bottom of the circle. Since the halfway down position calculations are complex and none of the provided positions match the given tension, we can conclude that the correct answer is: (D) none of the above