Simple Harmonic Motion | Lucent for GK - UPSC PDF Download

Understanding Periodic Motion

Periodic motion refers to the motion that repeats itself at regular intervals of time. To comprehend simple harmonic motion, it is essential to grasp the concept of periodic motion. We encounter numerous examples of periodic motion in our day-to-day lives. For instance, the swinging of a pendulum, the rocking of a cradle, the swaying of leaves due to a gentle breeze, and even the oscillation of a swing. In each of these examples, a particle performs a set of repetitive movements known as oscillation, which is a form of periodic motion. Simple Harmonic Motion serves as a prime example of oscillatory motion.

Exploring Simple Harmonic Motion

Definition of Simple Harmonic Motion
Simple Harmonic Motion refers to the motion in which an object moves back and forth along a straight line. Consider a pendulum as an example. When we swing a pendulum, it moves to and fro along the same line. These back-and-forth movements are called oscillations, making the oscillation of a pendulum a prime instance of simple harmonic motion. Additionally, imagine a spring fixed at one end. In its equilibrium position, when no external force is applied, the spring remains at rest. However, if we pull or push the spring, it exerts a force directed towards its equilibrium position. This force is known as the restoring force. Mathematically, the restoring force (F) is represented as F = -kx, where x represents the displacement of the spring from its equilibrium position, and k is the force constant. The negative sign indicates that the force acts in the opposite direction.

Linear Simple Harmonic Motion

Linear Simple Harmonic Motion refers to the linear periodic motion of a body, wherein the restoring force always points towards the equilibrium or mean position. The magnitude of this force is directly proportional to the displacement from the equilibrium position. It's important to note that while all simple harmonic motions are periodic, not all periodic motions are simple harmonic motions. To understand this concept further, let's revisit the previous example of a spring. Assuming the mass of the string is denoted as 'm,' the acceleration of the body (a) can be calculated as a = F/m = -kx/m = -ω^2x. Here, k/m is equivalent to ω^2, where ω represents the angular frequency of the body.

Concepts of Simple Harmonic Motion (S.H.M.)

To deepen our understanding of simple harmonic motion, let's explore a few fundamental concepts associated with it:

Amplitude

The amplitude of a particle represents the maximum displacement from its equilibrium or mean position. Its direction is always away from the mean or equilibrium position. The SI unit of amplitude is meters, and its dimensions are [L¹M⁰T⁰].

Period

The period of a particle is defined as the time taken to complete one full oscillation. In the case of simple harmonic motion, the period represents the minimum time required for the motion to repeat itself. Thus, the motion repeats itself after nT, where 'n' is an integer.

Frequency

Frequency refers to the number of oscillations performed by a particle in a given unit of time. The SI unit of frequency is hertz or rotations per second (r.p.s), with dimensions [L^0M^0T^-1].

Phase

The phase of simple harmonic motion signifies the state of oscillation. It is represented by the magnitude and direction of the particle's displacement. The initial phase of the motion is known as the epoch (α).

Difference Between Periodic Motion and Simple Harmonic Motion

It is essential to understand the distinctions between periodic motion and simple harmonic motion:

Periodic Motion:

• The displacement of the object in periodic motion may or may not align with the direction of the restoring force.
• Periodic motion can be oscillatory or non-oscillatory, depending on the scenario.
• Examples of periodic motion include the motion of clock hands or car wheels.

Simple Harmonic Motion:

• In simple harmonic motion, the displacement of the object always opposes the direction of the restoring force.
• Simple harmonic motion is inherently oscillatory.
• Examples of simple harmonic motion include the swinging of a pendulum or the motion of a spring.