NEET(UG)-2024 – Physics Questions with Solutions :Q1 -A bob is whirled in a horizontal plane by means of a string with an initial speed of w rpm
1. A bob is whirled in a horizontal plane by means of a string with an initial speed of w rpm. The tension in the string is T. If speed becomes 2w while keeping the same radius, the tension in the string becomes:
(1) T
(2) 4T
(3) T/4
(4)√ 2T
Answer is (2)
In the given scenario, the tension in the string will increase significantly when the speed of the bob doubles to 2w. Here’s why:
The tension in the string acts as the centripetal force, pulling the bob inwards and keeping it moving in a circular path. This centripetal force is related to the mass of the bob (m), its speed (v), and the radius of the circular path (r) by the following equation:
T = m * v^2 / r
where:
- T is the tension in the string
- m is the mass of the bob
- v is the speed of the bob
- r is the radius of the circular path
Since the mass (m) and radius (r) remain constant, the tension (T) depends solely on the square of the speed (v^2).
When the speed of the bob doubles from w to 2w, its squared speed increases by a factor of four:
- Initial squared speed: (w)^2
- Final squared speed: (2w)^2 = 4 * (w)^2
Therefore, based on the equation, the tension in the string will also increase by a factor of four:
- New tension (T’) = T * (4 * (w)^2) / ((w)^2)
- New tension (T’) = 4 * T
In conclusion, when the speed of the bob doubles, the tension in the string increases fourfold.
The concept behind the problem
The concept behind the problem revolves around understanding centripetal force and its relation to circular motion. Here’s a breakdown:
- Centripetal Force: Imagine you’re swinging a ball on a string. The string pulls the ball inwards, constantly changing its direction to keep it moving in a circle. This inward pull is called the centripetal force.
- Circular Motion: Objects moving in a circle experience a constant change in direction even though their speed might stay the same. This change in direction requires a force acting inwards, which is the centripetal force.
- Factors Affecting Centripetal Force: The strength of the centripetal force depends on two main factors:
- Mass (m) of the object: A heavier object requires a greater inward pull to keep it circling at the same speed.
- Speed (v) of the object: A faster-moving object needs a stronger inward pull to maintain its circular path.
- Mathematical Relationship: The centripetal force is calculated using the following equation:
- T = m * v^2 / r
- T: Tension in the string (acting as the centripetal force)
- m: Mass of the bob
- v: Speed of the bob
- r: Radius of the circular path (length of the string)
- Problem Application: In the given scenario, the mass (m) and radius (r) of the circular path remain constant. The problem asks how the tension (T) changes when the speed (v) doubles. Since the tension depends on the square of the speed (v^2), doubling the speed increases the tension by a factor of four.
This problem highlights how a seemingly simple change in speed (v) can significantly impact the centripetal force (T) required for circular motion.
