Contents
The laws of Physics Topics are used to explain everything from the smallest subatomic particles to the largest galaxies.
What is the Acceleration of a Particle in Simple Harmonic Motion?
Periodic Functions
A function f(t) is periodic if the function repeats itself after a regular interval of the independent variable t.
The simplest examples of a periodic motion can be represented by any of the following functions,
f(t) = Acos\(\frac{2 \pi}{T}\)t ……….. (1)
and g(t) = Asin\(\frac{2 \pi}{T}\)t ……… (2)
Here T is the time period of the periodic motion.
To check the periodicity of these functions t is to be replaced by (t + T). In the above equations simultaneously.
Hence equation (1) gives us,
f(t + T) = Acos[(\(\frac{2 \pi}{T}\)(t + T)] = Acos[\(\frac{2 \pi}{T}\)t + 2π]
= Acos\(\frac{2 \pi}{T}\)t [∵ cos(θ + 2π) = cosθ]
= f(t)
And equation (2) gives us,
g(t + T) = Asin[\(\frac{2 \pi}{T}\)(t + T)] = Asin(\(\frac{2 \pi}{T}\) + 2π)
= Asin\(\frac{2 \pi}{T}\)t [∵ sin(θ + 2π) = sinθ]
= g(t)
So, far a periodic function with period T,
f(t + T) = f(t)
or g(t + T) = g(t)
The result will be the same if we consider a linear combination of sine and cosine functions of period T,
i.e., f(t) = Asin\(\frac{2 \pi}{T}\)t + Bcos\(\frac{2 \pi}{T}\)t
Another example of periodic function is,
f(t) = sinωt + cos2ωt + cos4ωt
But, f(t) = e-ωt is not a periodic function, because it decreases monotonically with the increase in time and tends to zero as t → ∞
Displacement as a function of time: Displacement can be represented by a mathematical function of time. In case of periodic motion this function is periodic in nature. One of the simple periodic function is,
f(t) = Acosωt
When the argument ωt, is increased by an integral multiple of 2π radians, the value of the function remains the same.
Properties of Simple Harmonic Motion or SHM
Simple harmonic motion is the simplest form of oscillation. From the properties of simple harmonic motion, we can analyse any complex oscillation or vibration. Any type of oscillatory motion can be considered to be the resultant of two or more simple harmonic motions acting on a particle. Thus, it is of great importance to discuss SHM in detail.
Restoring force: When a vibrating particle is at its position of equilibrium, the resultant force acting on it is zero [Fig.]. For example, when a simple pendulum is at its equilibrium position the downward force due to the weight of the bob is balanced by the upward tension of the string.
So, the resultant force acting on the bob is zero. If the bob is displaced slightly from its equilibrium position and released, then a resultant force acts on the bob which tries to bring it to its equilibrium position. This force is called the restoring force. Since force is a vector quantity this restoring force has a magnitude and a direction. If the magnitude and the direction of the restoring force satisfy the following two conditions, the motion of the particle is termed simple harmonic motion (SHM).
- The restoring force is always directed towards the position of equilibrium of the particle.
- The magnitude of the restoring force is proportional to the displacement of the particle from its position of equilibrium.
Equation of Simple Harmonic Motion: Suppose, a particle is executing linear periodic motion along x -axis and the point O, which is the origin (x = 0), is the position of equilibrium of the particle [Fig.]. Let D be any point on the path of the particle with position coordinate x.
According to condition (ii), if F is the restoring force acting on the particle at D, then
F ∝ x
Again from condition (i), as the restoring force F is directed towards the equilibrium position, it is taken as negative since the displacement OD = x is taken as positive.
So, F = -kx …….. (1)
k is called the force constant and it is positive. Therefore, the magnitude of the restoring force acting on the particle when it is at a position of unit displacement is called the force constant. The unit of k are dyn ᐧ cm-1 (CGS) and N ᐧ m-1 (SI).
If m is the mass of the particle and a is its acceleration then F = ma. So, from equation (1) we get,
ma = -kx
or, a = –\(\frac{k}{m}\)x = -ω2x ……… (2)
Here, ω = + \(\sqrt{\frac{k}{m}}\) = constant ………. (3)
Any one of the equations (1) or (2) is called the equation of simple harmonic motion. As the forms of these equations are identical, it can be said that the properties of acceleration of the particle and those of the restoring force are identical. Simple harmonic motion can be defined with reference to the properties of acceleration.
Definition: The motion of a particle is said to be simple harmonic if its acceleration
- is proportional to its displacement from the position of equilibrium and
- is always directed towards that position.
It is to be noted that acceleration of a particle executing simple harmonic motion is expressed as a = -ω2x. Conversely, if the acceleration of a particle obeys the equation a = -ω2x, then we can say that the motion of the particle is simple harmonic.