Problem 19 - Master Resource Book in JEE Main Physics

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

(b) at the bottom of the circle

Step 1: Understand the Forces at the Top of the Circle

When the stone is at the top of the circle, both the gravitational force and the tension act downwards towards the center. The gravitational force is calculated using the formula \( F_g = m \cdot g \), where \( m = 2 \mathrm{~kg} \) and \( g = 9.8 \mathrm{~m/s^2} \). Therefore, \( F_g = 2 \cdot 9.8 = 19.6 \mathrm{~N} \). The centripetal force required is given by \( F_c = \frac{m \cdot v^2}{r} \), where \( v = 4 \mathrm{~m/s} \) and \( r = 1 \mathrm{~m} \). Thus, \( F_c = \frac{2 \cdot 4^2}{1} = 32 \mathrm{~N} \). The tension in the string, \( T \), is given by the equation: \( T = F_c - F_g = 32 - 19.6 = 12.4 \mathrm{~N} \).

Step 2: Understand the Forces at the Bottom of the Circle

When the stone is at the bottom of the circle, the tension in the string and the gravitational force both act upwards. Using the same centripetal force calculation \( F_c = 32 \mathrm{~N} \), the tension at the bottom is calculated by \( T = F_c + F_g = 32 + 19.6 = 51.6 \mathrm{~N} \).

Step 3: Analyzing Halfway Down the Circle

"Halfway down" refers to the side positions where the stone has no vertical component of gravitational force acting along the string's direction. Here, the centripetal force is the same by geometry, but it's not directly a condition where we calculate. Hence, it's less straightforward to discuss halfway tensions without specific computations or assumptions. However, we mainly assume specific circuit points unless stated otherwise in typical fundamental studies.

Step 4: Conclusion

From the calculations, the tension in the string is closest to \(52\mathrm{~N}\) at the bottom of the circle (\(51.6\mathrm{~N}\)). Thus, (b) is our answer.

Key Concepts

Centripetal Force

Centripetal force is a crucial concept in circular motion. It is the force required to make an object follow a circular path. The force always points towards the center of the circle. This force keeps the object moving in a circle rather than flying off in a straight line.
To calculate centripetal force, you use the formula:

• \( F_c = \frac{m \cdot v^2}{r} \)

Here, \( m \) is the mass of the object, \( v \) is the velocity, and \( r \) is the radius of the circle.
In our case, the stone’s centripetal force is calculated using these values: mass \( m = 2 \mathrm{~kg} \), velocity \( v = 4 \mathrm{~m/s} \), and radius \( r = 1 \mathrm{~m} \). So, \( F_c = \frac{2 \cdot 4^2}{1} = 32 \mathrm{~N} \).
This force is necessary at every point in the stone's path to maintain the circular motion.

Gravitational Force

Gravitational force is the force of attraction between two masses. On Earth, it gives weight to physical objects and acts downwards towards the center of the Earth. It is one of the forces at play when an object moves in a vertical circle.
The gravitational force on the stone is calculated by multiplying its mass by the gravitational acceleration \( g \), which is approximately \( 9.8 \mathrm{~m/s^2} \).
In case of our stone:

• The gravitational force \( F_g = m \cdot g = 2 \mathrm{~kg} \times 9.8 \mathrm{~m/s^2} = 19.6 \mathrm{~N} \)

When the stone is at the top of the circle, gravity assists the centripetal force, pulling it downwards. However, at the bottom, gravity acts adversely to the tension in the string, indicating why tension is higher at the bottom than at the top.

Tension in a String

Tension is the pulling force transmitted along a string, cable, or chain when it is pulled tight by forces acting from opposite ends. In the case of the stone moving in a vertical circle, tension in the string changes depending on its position in the circle.
At the top of the circle:

• Both the centripetal force and gravitational force act towards the center. The tension needed is less because gravity contributes to the centripetal force.
• The formula used here is \( T = F_c - F_g = 32 - 19.6 = 12.4 \mathrm{~N} \).

At the bottom of the circle:

• The tension is higher because it has to support the gravitational force in addition to providing the required centripetal force.
• Hence, \( T = F_c + F_g = 32 + 19.6 = 51.6 \mathrm{~N} \).

This tension is what keeps the stone moving in its circular path. If tension is not adequately calculated, the stone may not maintain its path in the circle correctly, leading to potential breakage or deviation from circular motion.