A 0.1 kg mass is suspended from a wire of negligible mass. The length of the wire is 1m and its cross-sectional area is 4.9 × 10–7 m2. If the mass is pulled a little in the vertically downward direction and released, it performs simple harmonic motion of angular frequency 140 rad s–1. If the Young's modulus of the material of the wire is n × 109 Nm–2, the value of n is
n = 4
This problem involves finding the Young's modulus of a wire based on the angular frequency of simple harmonic motion (SHM) when a mass is suspended from it. The key concept is that the wire acts like a spring, and its restoring force leads to SHM.
Step 1: Identify the System as a Spring-Mass System
When the mass is pulled down and released, the wire provides a restoring force proportional to the displacement, similar to a spring. Therefore, the motion is simple harmonic. For a spring-mass system, the angular frequency ω is given by:
where k is the spring constant and m is the mass.
Step 2: Relate the Spring Constant to Young's Modulus
For a wire under tension, the effective spring constant k is related to Young's modulus Y. When a wire of length L and cross-sectional area A is stretched, the force F required for an extension ΔL is given by Hooke's law for the wire:
Comparing this with the spring force F = k ΔL, we get the spring constant:
Step 3: Substitute into the Angular Frequency Formula
Substitute the expression for k into the angular frequency formula:
Square both sides to solve for Y:
Step 4: Plug in the Given Values
Given: m = 0.1 kg, L = 1 m, A = 4.9 × 10–7 m², ω = 140 rad/s. Substitute these into the equation:
Calculate step by step:
ω² = (140)² = 19600
ω² × m = 19600 × 0.1 = 1960
Numerator = 1960 × L = 1960 × 1 = 1960
So, Y = 1960 / (4.9 × 10–7) = (1960 / 4.9) × 107
1960 / 4.9 = 400
Therefore, Y = 400 × 107 = 4 × 109 N/m²
This matches the form n × 109, so n = 4.
Final Answer: n = 4
Simple Harmonic Motion (SHM): Periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. Examples include spring-mass systems, pendulums.
Elasticity and Young's Modulus: Young's modulus (Y) is a measure of the stiffness of a material, defined as the ratio of tensile stress to tensile strain. It is a property of the material and is constant for a given substance.
Spring Constant: For a spring, the spring constant k is the force required per unit extension. For other elastic bodies like wires, an effective spring constant can be derived based on their dimensions and material properties.
Angular frequency of spring-mass system:
Spring constant for a wire:
Young's modulus from SHM parameters:
A wooden wheel of radius R is made of two semicircular parts (see figure). The two parts are held together by a ring made of a metal strip of cross sectional area S and length L. L is slightly less than 2πR. To fit the ring on the wheel, it is heated so that its temperature rises by ΔT and it just steps over the wheel. As it cools down to surrounding temperature, it presses the semicircular parts together. If the coefficient of linear expansion of the metal is α, and its Young's modulus is Y, the force that one part of the wheel applies on the other part is :
