2 Most FAQ’s of Oscillations Chapter in Inter 1st Year Physics (TS/AP)

8 Marks

LAQ-1 : Show that the motion of a simple pendulum is simple harmonic and hence derive an equation for its time period. What is a seconds pendulum?

Introduction

A simple pendulum consists of a small mass, called a bob, suspended from a fixed point by a string of negligible mass and assumed to be inextensible. The motion of a simple pendulum is a classic example of Simple Harmonic Motion (SHM) under the influence of gravitational force.

Demonstration of SHM in a Simple Pendulum

1. For small angles (θ), the restoring force acting on the pendulum bob is proportional to the displacement from its equilibrium position. This force is due to gravity.
2. The restoring force (F) is given by F = −mg sin(θ), where m is the mass of the bob, g is the acceleration due to gravity, and θ is the angular displacement.
3. For small angles, sin(θ) can be approximated as θ (in radians), making the force F ≈ −mgθ.
4. This force being directly proportional to the negative of the displacement indicates that the motion of the pendulum is Simple Harmonic.

Derivation of Time Period of a Simple Pendulum

1. The equation of motion for the pendulum can be written as $$m\frac{d^2x}{dt^2} = -mgθ$$ simplifying to $$\frac{d^2θ}{dt^2} + \frac{g}{l}θ = 0$$ where l is the length of the pendulum.
2. Comparing this with the standard SHM equation $$\frac{d^2x}{dt^2} + ω^2x = 0$$ we find $$ω^2 = \frac{g}{l}$$
3. The time period (T) of SHM is given by $$T = 2π \sqrt{\frac{1}{ω^2}}$$ which leads to the formula $$T = 2π \sqrt{\frac{l}{g}}$$​​ for the time period of a simple pendulum.

Concept of a Seconds Pendulum

1. A Seconds Pendulum is a simple pendulum whose time period for one complete oscillation is exactly two seconds, one second for a swing in one direction and one second for the return swing.
2. To calculate the length of a seconds pendulum, use the formula $$T = 2π \sqrt{\frac{l}{g}}$$ with T set to 2 seconds. Solving for l gives the necessary length of the pendulum for it to be a seconds pendulum.

Summary

The motion of a simple pendulum is Simple Harmonic Motion for small angular displacements. The time period of a simple pendulum is derived as $$T = 2π \sqrt{\frac{l}{g}}$$ A Seconds Pendulum specifically refers to a simple pendulum with a time period of exactly two seconds, a historically significant timekeeping standard.

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LAQ-2 : Define simple harmonic motion. Show that the motion of projection of a particle performing uniform circular motion, on any diameter is simple harmonic.

Definition of Simple Harmonic Motion

Simple Harmonic Motion (SHM): SHM is a type of periodic motion where the force acting on the moving object is directly proportional to the displacement from its equilibrium position and is always directed towards that position. This motion is characterized by a sinusoidal oscillation about an equilibrium point.

SHM as a Projection of Uniform Circular Motion

The concept of SHM can be visualized through the motion of a particle performing uniform circular motion (UCM) when projected onto any of its diameters.

1. Uniform Circular Motion and Its Projection:
   • Consider a particle performing UCM in a circle of radius r with a constant angular velocity ω.
   • The projection of this motion onto any diameter of the circle results in oscillatory motion back and forth along that diameter.
2. Analyzing the Projection:
   • At any time t, the particle’s position in the circular path can be described by an angle θ = ωt.
   • The projection on the diameter (let’s say the x-axis) will have a displacement
     x = r cos(θ) = r cos (ωt).
   • This equation x = r cos(ωt) represents a sinusoidal motion, which is the hallmark of SHM.
3. Proof of SHM:
   • The restoring force in SHM is given by F = −kx, where k is the force constant.
   • For the projected motion, the restoring force is the component of the centripetal force along the diameter, which is proportional to the displacement x.
   • This proportional relationship between restoring force and displacement, and the sinusoidal nature of the motion, confirms that the projection of UCM on any diameter is indeed Simple Harmonic.

Summary

Simple Harmonic Motion is defined by the sinusoidal oscillation of an object about an equilibrium point with a restoring force proportional to the displacement. The projection of a particle in uniform circular motion onto any of its diameters illustrates an example of SHM, where the motion along the diameter mirrors the characteristics of SHM. This relationship provides a foundational understanding of SHM through the lens of circular motion.