Problem 24 - Master Resource Book in JEE Main Physics

Question

A uniform chain of length \(L\) and mass \(M\) is lying on a smooth table and one third of its length is hanging vertically down over the edge of the table. If \(g\) is acceleration due to gravity, the work required to pull the hanging part on the the table is (a) \(M g L\) (b) \(\mathrm{Mgl} / 3\) (c) \(\mathrm{Mg} \mathrm{L} / 9\) (d) \(\mathrm{Mgl} / 18\)

Step-by-Step Solution

Answer

The work required is \( \frac{MgL}{18} \), option (d).

Step 1: Determine the Length of the Hanging Chain

The problem states that one-third of the chain's length is hanging over the edge. So if the total length of the chain is \( L \), the hanging part of the chain has a length of \( \frac{L}{3} \).

Step 2: Calculate the Mass of the Hanging Part

Since the chain is uniform, its mass per unit length is \( \frac{M}{L} \). Therefore, the mass of the hanging part is given by \( \frac{M}{L} \times \frac{L}{3} = \frac{M}{3} \).

Step 3: Determine the Gravitational Force on the Hanging Part

The gravitational force acting on the hanging part of the chain is given by \( \text{mass} \times g = \frac{M}{3} \times g = \frac{Mg}{3} \).

Step 4: Calculate the Work Done to Pull the Hanging Part onto the Table

To pull the chain onto the table, we need to do work against gravity. The center of mass of the hanging part is at a height \( \frac{1}{2} \times \frac{L}{3} = \frac{L}{6} \) below the table. The work done is \( ext{force} \times ext{distance} = \frac{Mg}{3} \times \frac{L}{6} = \frac{MgL}{18} \).

Key Concepts

Uniform Chain Problem

The uniform chain problem is a popular exercise in mechanics that helps illustrate concepts of work and energy. In this problem, we often deal with a chain that is draped over an edge, such as a table. The chain is uniform, meaning its mass is evenly distributed along its length. This assumption simplifies calculations:

• The mass per unit length of the chain is constant.
• Sections of the chain can be considered separately while maintaining proportional relationships of mass and length.

For example, if the total length of the chain is given as \( L \) and the total mass \( M \), any section of the chain encompassing a fraction of \( L \) will have a corresponding fraction of \( M \). In problems involving such a chain, it’s crucial to:

• Determine the length of the hanging part.
• Calculate the corresponding mass of the hanging section.
• Use these to find other physical quantities like forces or work done.

These principles will guide you through similar problems in mechanics, where equilibrium, gravity, or tension play critical roles.

Gravitational Force

Gravitational force is a fundamental concept in physics, describing the attraction between two bodies with mass. In the uniform chain problem, this force is what pulls the hanging part of the chain downward.For any mass \( m \) close to the Earth's surface, gravitational force can be calculated by:\[ F = m \times g \]Where:

• \( F \) is the gravitational force.
• \( m \) is the mass.
• \( g \) is the acceleration due to gravity, approximately equal to \( 9.8 \text{ m/s}^2 \).

In the case of the hanging chain, you need to calculate the force by substituting the mass of the hanging section. Understanding gravitational force is key to solving problems involving weight and motion, allowing us to calculate the work needed to counteract these forces to move objects.

Center of Mass

Center of mass is an essential concept when discussing the work done on objects. It is the point where the mass of an object is considered to be concentrated, affecting how we compute forces and resultant movements. For the hanging part of the uniform chain:

• The center of mass will lie halfway along its length.
• If one-third of the chain is hanging, its center of mass is at \( \frac{L}{6} \) below the table's edge.

This point is crucial when calculating work done because the work required to elevate the chain is computed based on the movement needed at the center of mass. Knowing the center of mass provides insight into:

• Balance and stability of objects.
• The effort needed to lift or move sections of an object.

This concept is applicable not just to chains but to a broad range of physics problems, particularly where rotation or translation about a point is involved.

Kinematics

Kinematics is the branch of mechanics that deals with the motion of objects without considering the forces causing the motion. In the context of pulling the chain back onto the table, we are interested in how the chain's position changes over time. While solving this problem, we implicitly use kinematics by examining how the height (or position) of the center of mass changes as the chain is lifted. For this motion:

• Initial and final positions of the chain are considered.
• Movement occurs vertically against gravity.

Understanding kinematics involves:

• Grasping concepts of displacement, velocity, and acceleration.
• Analyzing how an object's position changes over time.

Though not directly calculated in this exercise, kinematic principles ensure a structured approach to understanding the motion involved when lifting the chain.

Accelaration due to Gravity

Acceleration due to gravity is a constant that greatly simplifies solving problems involving mass and force near the Earth's surface. It's the rate at which an object's velocity changes due to Earth's gravitational pull, denoted by \( g \).For problems like the uniform chain, this constant allows us to relate gravitational force to mass:

• Typically valued at \( 9.8 \text{ m/s}^2 \).
• Affects how we calculate other forces and work done due to gravity.

Acceleration due to gravity is utilized in calculating the gravitational force and resulting work done:

• It’s independent of the object's mass and only depends on its interaction with the Earth.
• Simplifies equations when estimating motion under gravity.

This concept is fundamental not just in textbook exercises but also in a multitude of real-world applications, from determining the trajectory of falling objects to engineering calculations that ensure the structural feasibility of buildings and bridges.