Question

A rubber cord catapult has cross-sectional area \(25 \mathrm{~mm}^{2}\) and initial length of cord is \(10 \mathrm{~cm} .\) It is stretched to \(5 \mathrm{~cm}\) and then released to project a missile of mass \(5 \mathrm{gm}\). Taking \(\mathrm{Y}_{\text {rubber }}=5 \times 10^{8}\left(\mathrm{~N} / \mathrm{m}^{2}\right)\) velocity of projected missile is \(\ldots \ldots .\) (A) \(20(\mathrm{~m} / \mathrm{s})\) (B) \(100(\mathrm{~m} / \mathrm{s})\) (C) \(250(\mathrm{~m} / \mathrm{s})\) (D) \(200(\mathrm{~m} / \mathrm{s})\)

Short answer

The velocity of the projected missile is \(20 \mathrm{~m} / \mathrm{s}\) (A).

Step-by-step solution

Calculating the stress in the rubber cord

The stress in the rubber cord can be calculated using Hooke's law, which states that the stress is directly proportional to the strain: Stress \(= Y \times \) Strain where Y is Young's modulus. First, let's find the strain in the rubber cord when it is stretched by 5 cm. Strain \(= \frac{\Delta L}{L}\), where \(\Delta L\) is the change in length, and L is the initial length. Strain = \(\frac{5 \mathrm{~cm}}{10 \mathrm{~cm}} = \frac{1}{2}\) Now, we can find the stress in the rubber cord: Stress \(= Y \times\) Strain = \(5 \times 10^8 \times \frac{1}{2}\) N/m².

Finding the Force applied by the rubber cord

We can find the force applied by the rubber cord using the formula: Force = Stress × Cross-sectional Area Cross-sectional Area = \(25 \mathrm{~mm}^{2}\) = \(25 \times 10^{-6} \mathrm{m}^{2}\) Now, let's find the force: Force = \(5 \times 10^8 \times \frac{1}{2} \times 25 \times 10^{-6}\) N.

Calculating the work done by the rubber cord

To calculate the work done by the rubber cord, we need to find the potential energy stored in it when it is stretched. The formula for the potential energy stored in a stretched linear elastic material is: Potential Energy \(= \frac{1}{2} \times\) Force × Change in Length Substituting the values we derived earlier, we can find the potential energy: Potential Energy \(= \frac{1}{2} \times (5 \times 10^8 \times \frac{1}{2} \times 25 \times 10^{-6}) \times 5 \times 10^{-2}\) J.

Calculating the velocity of the projected missile

According to the conservation of energy principle, the potential energy stored in the stretched rubber cord is completely transferred to the kinetic energy of the missile. So, the kinetic energy of the missile can be calculated as: Kinetic Energy \(= \frac{1}{2} \times\) Mass × Velocity². We know the mass of the missile is 5 gm = \(5 \times 10^{-3}\) kg. Now, equating the potential energy and kinetic energy: \(\frac{1}{2} \times (5 \times 10^8 \times \frac{1}{2} \times 25 \times 10^{-6}) \times 5 \times 10^{-2} = \frac{1}{2} \times 5 \times 10^{-3} \times\) Velocity² Solving for the velocity: Velocity = 20 m/s So, the velocity of the projected missile is 20 m/s, which corresponds to option (A).

Key concepts

Elastic Potential Energy

Elastic potential energy refers to the energy stored in an object when it is stretched or compressed. This is seen in objects like rubber bands, springs, and elastic cords. When a rubber cord is stretched, it stores potential energy due to the work done to pull it apart.
This energy is computed using the formula: Potential Energy \(= \frac{1}{2} \times \text{Force} \times \text{Change in Length}\).
Here, the force is generated due to the elasticity of the material. The stored energy can then be released to do work, as in the case of a catapult launching a projectile.
Elastic potential energy becomes prominently valuable because it allows the transformation of stored energy into kinetic energy, enabling objects to move. This concept is integral in understanding systems like spring-powered devices, where the stored energy directly influences the motion and speed of an object.
When you calculate it, ensure you know the force applied and the extent of elongation or compression to get an accurate measure of the energy stored in the elastic medium. This energy transformation principle is crucial in various practical applications.

Young's Modulus

Young's Modulus, often denoted by \(Y\), is a measure of the stiffness of a material. It tells us how much a material will stretch or compress under a particular stress. The formula used is:
\[ Y = \frac{\text{Stress}}{\text{Strain}} \] Here’s a breakdown of the terms:

• Stress: When a force is applied to a material, the internal forces between molecules distribute this force throughout the object. Stress is the force applied per unit area, usually measured in Newtons per square meter (N/m²).
• Strain: This measures how much a material deforms under stress. It is the ratio of change in length to the original length.

Young's Modulus is especially crucial in materials science and engineering because it indicates how a material will react to forces applied to it. A higher modulus means the material is stiffer and less prone to deformation.
In practical applications, knowing the Young's Modulus helps engineers and designers choose the right materials for specific uses, ensuring durability and performance. For instance, in the rubber cord of a catapult, Young's Modulus helps predict how far it will stretch and how much energy it can store before releasing a projectile.

Conservation of Energy

The principle of conservation of energy is a fundamental concept in physics. It states that energy cannot be created or destroyed but only transformed from one form to another. In the context of elastic bodies like rubber cords, this principle is crucial to understand how potential energy is converted into kinetic energy.
When a rubber cord is stretched, it holds elastic potential energy. Upon release, this potential energy is converted into kinetic energy, which propels the missile forward. Therefore, the total energy before and after the transformation remains constant, assuming no losses to air resistance or heat.
The formula governing the conservation of energy in this scenario is:

• Initially, Potential Energy: Measured when the rubber is stretched.
• Finally, Kinetic Energy: Measured when the object is in motion, given by \( \frac{1}{2} \times \text{Mass} \times \text{Velocity}^2 \).

By equating the elastic potential energy to the kinetic energy of the missile, we can predict the velocity of the missile once the energy transfers. This principle not only simplifies complex physical interactions but also facilitates the development of energy-efficient systems and devices across multiple fields. Understanding energy conservation is key to unlocking many scientific and engineering problems.