Simple Harmonic Motion Equation

A motion that is repeated by itself after regular intervals of time is called periodic motion. When a body is in an equilibrium position, no resultant external force acts on it. At this position, if a body is at rest, it will stay at rest until disturbed by an external force. If a small displacement is given to the body, a force acts on the body, trying to return the body to the original position of the equilibrium point. This gives rise to oscillations or vibrations. Examples of oscillating motion are a simple pendulum and when a ball placed at the bottom of a bowl is given a displacement.

Simple Harmonic Motion:

Simple harmonic motion is a type of periodic motion around a central or equilibrium point, such that the maximum displacement on either side of the central position is equal. The force acting on such a body always points towards the equilibrium point and is directly proportional to the distance of the body from the equilibrium point i.e. 

F = −kx, 

where F = force, x = displacement of the body, and k = a constant. This equation is known as Hooke’s law. 

An example of a simple harmonic oscillation is the vibration is a mass attached to a hanging spring, with one end of the string fixed at the hanging point. When the spring is stretched with the maximum displacement, say −x, it is having maximum tension, which pulls the mass upwards. When the spring is compressed with the maximum displacement +x, it has its maximum compression force, which pushes the mass downwards again. 

At either position of maximum displacement, the force is maximum and points towards the equilibrium or central point. At the maximum displacement point, the body’s velocity is zero, but its acceleration is maximum, and the direction of the body reverses. 

At the equilibrium position, the body’s velocity becomes maximum, but the acceleration becomes zero. There are many systems that demonstrate simple harmonic motion, such as an oscillating pendulum, vibrating particles in a sound wave and electrons carrying alternating current in a wire, etc.

Simple Harmonic Motion Equation:

Simple harmonic motion (SHM) is a periodic motion in which the displacement is a sinusoidal function of time. The sinusoidal equation can be derived by considering the illustration of a simple pendulum.

The vertical angle, as a function of time, can be considered a displacement variable in the case of a simple oscillating pendulum. Both negative and positive values are possible for the displacement variable. For experiments over oscillations, the displacement of the body is determined at different times. This displacement is shown by a mathematical time function. The periodic mathematical function can be represented as

f(t) = Xcosωt …(i)

If the ωt argument increases by any integral multiple of 2π, the value of the function f(t) does not change. Thus, f(t) is periodic and its time period, T can be represented as 

T = 2πω …(ii)

Hence, the function f (t) is periodic with time period T,

f(t) = f (t+T)

The same result holds true for sine function as well:  

f(t) = Xsinωt …(iii)

A linear combination of the sine function and the cosine functions is also a periodic function with the same time period T:

f(t) = Xsinωt + Ycosωt …(iv)

If we put the values X = Acosφ and Y = Asinφ in equation (iv): 

f(t) = Asin(ωt + φ)

Here A and φ are constants given by:

A = √X²+Y² 

Φ = tan-1Y/X

From the above SHM Equation, the formula for the time period of a simple pendulum can also be derived.

Let θ be the vertical angle made by the string of length L. The resultant force mg can be divided into the components mgcosθ and mgsinθ, along the string and perpendicular to it, respectively. The net radial force T –mg cosθ provides the radial acceleration, and mgsinθ provides the tangential acceleration. Torque τ about the support is provided by the tangential component of force

τ = –L(mgsinθ) 

By rotational motion law: 

τ = I α 

where I = moment of inertia of the system, α = angular acceleration. 

Hence, I α = –Lmgsinθ 

=> α = −mgLsinθ / I

=> α = −mgLθ / I

Since α(t) = –ω2x(t) 

=> ω= √mgL/I 

And hence,

T = 2π√ L/g 

Where T is the time period of a simple pendulum.

Conclusion:

Periodic motion is the motion that repeats itself after regular intervals of time. Simple Harmonic Motion is a type of periodic motion around a central or equilibrium point, such that the maximum displacement on either side of the central position is equal. The force acting on such a body always points towards the equilibrium point and is directly proportional to the distance of the body from the equilibrium point. The displacement of such a motion is a sinusoidal function of time. The displacement equation can be represented by a linear combination of sine and cosine functions as f(t) = Asin(ωt + φ). The article also includes examples of oscillating motion for guidance.