# Simple Harmonic Motion

â€‹We shall first discuss the general dynamics of an SHM and later analyse a few specific examples of it. We know that work has to be done on a system to displace it from its position of equilibrium. The restoring force F obviously depends on the work done to give a displacement x.

Thus, F is some general function of x. For systems oscillating violently (large x) the dependence of F on x is very complex. We shall, however, not deal with such systems but focus our attention on systems in which the moving part always stays close to its mean position (small x). This is called small oscillation approximation. For such systems, the restoring force is proportional to the displacement and opposes its increase. In other words,

F = - kx ----------(i)

The negative sign indicates that F opposes increase in x. K, the constant of proportionality, is called the force constant. The MKS unit for K is Nm^{-1}. The magnitude of K depends on the elastic properties of the system under study. In the specific examples of SHM we shall compute the value of K for each case. For example, in the system consisting of a mass and a spring, K will depend on the stiffness or strength of the spring.

The above equation (i) is a statement of HookeÃ¢â‚¬â„¢s law for elastic forces. The general definition of SHM is the motion in which the restoring force is proportional to the displacement from the mean position and opposes its increase. We shall see that in such a motion the displacement varies harmonically with time.

# Equation of Motion

Under the influence of a restoring force F = (-kx) a body acquires a velocity and hence an acceleration.

If m is the mass of the body, then from NewtonÃ¢â‚¬â„¢s law (force = mass acceleration), the acceleration is given by

-------- (ii)

The physical statement corresponding to this equation is that the acceleration is proportional (and opposite) to the displacement. In order to determine what type of motion is represented by this equation we need to solve this differential equation, i.e. obtain an expression of displacement x as a function of time.

# Solution to the Equation of Motion

The usual procedure to solve any differential equation is to guess a solution and see if it works. Equation (ii) relates a function x(t) (not yet known) to its second derivative.

To satisfy this equation we have to look for a function x(t) whose second derivative, except for a negative constant factor , is the same as the function x(t) itself. Our knowledge of calculus tells us that sine and cosine functions have just this property, since

and

One can immediately verify that functions (a sin Î¸) and (b cos Î¸) , where a and b are constants, also obey this property. Since angle Î¸, measured in radians, must depend on time t, we set Î¸ = Ï‰t where Ï‰ is a constant to be measured in radians/second.

Thus, let us try a solution for the equation

x(t) = a sin Ï‰ t

Differentiating twice with respect to t, we get

Substitution in equation (ii) gives

Therefore, if we choose the constant such that ----(iii)

Then x (t) = a sin Ï‰t, is indeed , a solution of the equation. We can similarly verify that x (t) = b cos Ï‰t is also a solution of equation.

So the displacement of the particle from a certain chosen origin is found to vary with time as:

X(t) = A Cos (Ï‰ t + Ï† )

where A is the amplitude, Ï† is the phase difference and Ï‰ is the angular frequency.

The quantity A is a positive constant which represents the maximum displacement of the particle or object in either direction.

The time varying quantity (Ï‰ t + Ï† ) is known as phase of the motion and it describes the state of motion at a given time.

If there is no phase difference then Ï† = 0, so the equation of simple harmonic motion can be written as

X(t) = A Sin Ï‰ t

Any periodic motion has to reach its initial position after one period of the motion i.e. x(t) must be equal to X(t+T) for all t.

A Sin Ï‰ t = A Sin Ï‰ (t+T)

The sine function repeats itself when its argument has increased by 2Ï€

Ï‰ (t+T) = Ï‰ t + 2Ï€

Ï‰ t + Ï‰ T = Ï‰ t + 2Ï€

Ï‰ T = 2Ï€

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i.e., the value of the function at time is equal to the value the function had at an earlier time t. In other words, the function repeats itself after a time interval 2Ï€ /Ï‰ . Therefore the period T is given by
Thus, we find that the constant Ï‰ is related to the period (T) of motion by Equation (vii). We would obtain the same result if we had used the cosine function [Equation (vi)] since
This shows that functions like sin Ï‰t and cos Ï‰t are periodic having a period T = 2Ï€ /Ï‰ .The constant f and d do not depend upon time and hence do not affect the periodicity. In terms of period T, these functions are sin and cos.**

Notice that these functions repeat also at time intervals which are multiples of T, i.e., at intervals 2T, 3T, Ã¢â‚¬Â¦ etc. But T is the smallest time interval after which these functions repeat themselves, i.e., the period of these functions is T.

# Meaning of Ï‰

The smallest time interval after which a motion repeats itself is called the time period (or simply period) of the motion. It is usually denoted by the symbol T. If the time t in Equation (v) is increased to a value , the function becomes

= A sin (Ï‰ t + Ï†)= x (t)

-------(vii)

Notice that these functions repeat also at time intervals which are multiples of T, i.e., at intervals 2T, 3T, Ã¢â‚¬Â¦ etc. But T is the smallest time interval after which these functions repeat themselves, i.e., the period of these functions is T.

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