Problem 104 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

A body weighing \(13 \mathrm{~kg}\) is suspended by two strings \(5 \mathrm{~m}\) and \(12 \mathrm{~m}\) long, their other ends being fastened to the extremities of a rod \(13 \mathrm{~m}\) long. If the rod be so held that the body hangs immediately below the middle point, then tensions in the strings are (a) \(5 \mathrm{~kg}\) and \(12 \mathrm{~kg}\) (b) \(5 \mathrm{~kg}\) and \(13 \mathrm{~kg}\) (c) \(12 \mathrm{~kg}\) and \(13 \mathrm{~kg}\) (d) \(5 \mathrm{~kg}\) and \(5 \mathrm{~kg}\)

Step-by-Step Solution

Answer

The tensions are 5 kg and 12 kg.

Step 1: Understand the Setup

We have a rod that is 13 meters long, and a body weighing 13 kg is suspended from the midpoint of this rod using two strings of lengths 5 m and 12 m. The entire system is in equilibrium.

Step 2: Identify Forces in Equilibrium

Since the rod is held so that the body hangs exactly below its midpoint, each string experiences a tension force. We will refer to the tensions in the strings as \( T_1 \) and \( T_2 \). The body weighs \( 13 \text{ kg} \), so the sum of the tensions \( T_1 + T_2 = 13 \text{ kg} \).

Step 3: Apply Pythagoras Theorem

The situation described is a classic right triangle problem. The lengths of the strings and the rod suggest a right triangle with sides 5 m, 12 m, and 13 m, a well-known Pythagorean triple.

Step 4: Relate Lengths to Tensions

In a system of equilibrium involving right triangles, if the rod is balanced equally length-wise (13 m) and the weights (strings) are given as 5 m and 12 m, then the tensions in the strings, \( T_1 \) and \( T_2 \), relate directly to these lengths. The tensions are proportional to the lengths of the respective sides of the triangle (due to the equilibrium condition set by the Pythagorean theorem).

Step 5: Conclusion on Tensions

Thus, \( T_1 \) for the 5 m string should be 5 kg, and \( T_2 \) for the 12 m string should be 12 kg. Verify that these add up to the 13 kg weight as required by equilibrium.

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that helps us determine the relationship between the sides of a right triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In mathematical terms, it is expressed as:

\[ c^2 = a^2 + b^2 \]Here, \( c \) is the hypotenuse, while \( a \) and \( b \) are the other two sides.

In the exercise, we have a triangle formed by the rod and the two strings. The Pythagorean triple \( 5, 12, 13 \) fits perfectly here:

• Rod (hypotenuse): 13 meters
• String 1: 5 meters
• String 2: 12 meters

This shows that these lengths satisfy the Pythagorean Theorem, creating a right triangle. Therefore, the configuration confirms a state of equilibrium, and helps us in understanding the force distribution.

Tension in Strings

Tension is a force that is transmitted through a string, rope, cable, or similar object when it is pulled tight by forces acting on opposite ends. In this scenario, the strings are under tension because they hold up a weight of 13 kg.

The suspending strings have different lengths, and this affects their respective tension forces. The force of tension is related to several factors:

• Weight of the object (13 kg in this case)
• Length of the strings (5 m and 12 m)
• The angle at which each string hangs.

Since the rod is perfectly balanced in the middle with the body hanging directly beneath it, each string experiences a tension that is directly proportional to its length:

• String of 5 meters exerts 5 kg of tension.
• String of 12 meters exerts 12 kg of tension.

This is because the force distribution due to equilibrium ensures that the sum of tensions equates to the total weight, thus maintaining balance.

Mechanics

Mechanics is the branch of physics that deals with the motion of objects and the forces that act upon them. In this exercise, we focus on statics — a sub-branch of mechanics concerning objects in equilibrium, where forces are balanced with no net motion.

For an object to be in equilibrium:

• All forces acting on the object must balance out so that the object does not move.
• In the case of the suspended body, the tensions in the strings perfectly counteract its weight.

By analyzing these mechanics, we realize that the problem is elegantly set up as a balanced force scenario. The rod's placement and the length of the strings ensure that we have a determined equilibrium state:

• The downward force due to the body's weight is balanced by the upward tension forces exerted by the strings.
• The angles formed and lengths of the strings allow the application of the Pythagorean Theorem to confirm balance.

Such mechanical problems help illustrate fundamental physics concepts, bridging them with geometric principles to portray the wonder of interconnected forces acting in harmony.