1.6 Composition of harmonic motions

Composition, here, means combining more than one simple harmonic motion. However, this statement needs to be interpreted carefully. Every particle has only one motion at a time. Actually, we mean to combine two or more harmonic motions, which result from the operation of forces, each of which is individually capable of producing SHM. Therefore, it is actually combining SHMs, produced by forces operating on a “single” particle.

Further, we need to emphasize one important limitation to our discussion. Our context is limited to combining effects of forces which produce SHMs of “same” frequency.

We organize our study under two heads :

• Composition of SHMs along same straight line.
• Composition of SHMs along two mutually perpendicular straight lines.

Force analysis

The motion of a particle, acted upon by two or more forces, is governed by Newton’s second law. The situation, here, is no different except that each of the forces is characterized for being proportional to negative of displacement. This means that each of the forces, if left to act alone, would produce SHM. Nonetheless, we have seen that force is a vector quantity with the underlying characteristic that each force produces its effect independent of other force.

The independence of a force to the presence of other forces makes our task easy to assess the net result. Let us consider the motion of a particle as if it is being worked alone by a single force. Let “ $r_1$ ” and “ $v_1$ ” be the position and velocity at a particular instant resulting from the action of this force. Now, let “ $r_2$ ” and “ $v_2$ ” be the position and velocity at a particular instant resulting from the action of the other force as if it is the only force working on the particle.

The resultant position and velocity vector, then, would simply be the vector additions of individual quantities. Hence, position and velocity of the particle, when operated simultaneously by two forces, would be :.

$$r=r_1+r_2$$

$$v=v_1+v_2$$

These equations provide the general framework for studying motion that result from action of more than one force capable of producing SHM.

Composition of shms along same straight line

Let us consider two SHM forces, $F_1$ and $F_2$ , acting along the same straight line. Let the displacements be given by two equations,

$$x_1=A_1\sin \omega t$$

$$x_2=A_2\sin \left(\omega t+\phi \right)$$

We have written two displacements which reflect a convenient general case. Amplitudes are different. At any given instant, one of the two SHMs is “ahead of” or “lags behind” other, depending on the sign of phase constant “φ”. As pointed out earlier, we have kept the angular frequency “ω” same for both SHMs.

Now, we want to find the net displacement of the particle at any given instant. Referring to our earlier discussion, we can find net displacement by evaluating vector relation,

$$r=r_1+r_2$$

Since both SHMs are along the same straight line, we can drop the vector sign and can simply write this relation in the present context as :

$$\Rightarrow x=x_1+x_2$$

$$x=A_1\sin \omega t+A_2\sin \left(\omega t+\phi \right)$$

Expanding trigonometric function,

$$\Rightarrow x=A_1\sin \omega t+A_2\sin \omega t\cos \phi +A_2\cos \omega t\sin \phi$$

Segregating sine and cosine functions, keeping in mind that “φ” is constant :