Problem 15 - Applied Mathematics

Question

A particle is hanging in equilibrium at one end of an elastic string whose other end is fixed. Find the distance between the particle and the fixed end: (a) the particle weighs \(10 \mathrm{~N}\), (b) the modulus of elasticity is \(8 \mathrm{~N}\), (c) the natural length of the string is \(2 \mathrm{~m}\).

Step-by-Step Solution

Answer

3.25 meters

Step 1 - Identify the forces acting on the particle

The two forces acting on the particle are its weight (downwards) and the elastic restoring force of the string (upwards). Since the particle is in equilibrium, these forces must be equal in magnitude.

Step 2 - Write the equilibrium condition

Let the extension of the string be denoted by \( e \). The restoring force by an elastic string is given by Hooke's Law: \[ F = k \times e \] where \( k \) is the modulus of elasticity. The weight of the particle \( w = 10 \mathrm{~N} \). Therefore, \[ 10 = 8 \times e \]

Step 3 - Solve for the extension

Solve the equation \[ 10 = 8 \times e \] to find the extension, \( e \): \[ e = \frac{10}{8} = 1.25 \mathrm{~m} \]

Step 4 - Calculate the total distance

The total distance between the particle and the fixed end is the sum of the natural length of the string and the extension. Given the natural length is 2 meters, total length is: \[ L = 2 + 1.25 = 3.25 \mathrm{~m} \]

Key Concepts

Hooke's Law

Hooke's Law is essential in understanding how elastic materials like strings and springs behave under forces. The law states that the force exerted by an elastic material is directly proportional to the amount of stretch or compression it experiences. Mathematically, the formula is expressed as: \[ F = k \times e \]
Here, \( F \) represents the force exerted, \( k \) is the modulus of elasticity (or spring constant), and \( e \) is the extension or compression. This law helps us calculate how much a string or spring will stretch when a force is applied. In the exercise above, Hooke's Law is used to find the extension of the string by setting the weight and restoring force equal, as the particle is in equilibrium.

Modulus of Elasticity

The modulus of elasticity, also known as the spring constant, indicates how stiff or flexible an elastic string or spring is. It is denoted by \( k \) in Hooke's Law. A larger value of \( k \) means the string is stiffer and would stretch less under the same force.
For example, in the given exercise, the modulus of elasticity is \( 8 \text{ N} \text{m}^{-1} \), meaning for every meter of stretch, the restoring force is \( 8 \text{ N} \). This concept is crucial when analyzing how materials will deform under various forces, which is a common problem in physics and engineering.

Natural Length of the String

The natural length of an elastic string is the length of the string when no external forces are acting on it, meaning it is neither stretched nor compressed. In our problem, the natural length is given as \( 2 \text{ m} \).
When a force is applied, the string will extend beyond its natural length. This natural length is the baseline measurement before any force causes a deformation. Knowing the natural length is vital because the total length when the string is stretched will be this natural length plus the additional extension.

Equilibrium of Forces

In the context of this problem, equilibrium of forces means that the weight of the particle pulling down is balanced by the elastic force pulling up. Because the particle is not moving, the net force acting on it is zero.
We set the downward gravitational force equal to the upward elastic force to find the extension of the string. The equilibrium condition can be written as: \[ w = F \]
Here, \( w \) is the weight (\( 10 \text{ N} \)) and \( F \) is the elastic force calculated using Hooke's Law.

Particle Mechanics

Particle mechanics involves studying the motion and equilibrium of particles under various forces. In our exercise, the particle is influenced by the gravitational force (its weight) and the elastic force from the string.
Understanding these forces and how they interact helps us determine the system's behavior—like the particle's position in equilibrium. These principles are not just limited to strings but can be applied to various scenarios in physics and engineering, such as studying the motion of objects, stability of structures, and more.