Question

Three simple harmonic motions in the same direction having the same amplitude $A$ and same period are superposed. If each differ in phase from the next by $60^\circ$, then choose the correct statement(s).

• A. resultant amplitude is 2A
• B. phase of the resultant motion relative to the motion having least phase is $60^\circ$
• C. energy associated with the resulting motion is 4 times the energy associated with any single motion
• D. resulting motion is not simple harmonic

Answer 1

Answer: (A), (B), (C)

Resultant amplitude is 2A, phase of the resultant motion relative to the motion having least phase is $60^\circ$, energy associated with the resulting motion is 4 times the energy associated with any single motion

Answer 2

Let the three SHMs be:

$$x_1=A\cos\omega t, \quad x_2=A\cos(\omega t+60^\circ), \quad x_3=A\cos(\omega t+120^\circ).$$

Representing them as phasors:

• $\vec{A}_1 = A\angle 0^\circ$
• $\vec{A}_2 = A\angle 60^\circ$
• $\vec{A}_3 = A\angle 120^\circ$

Step 1: Add the phasors

Break into components:

$$\begin{aligned} \vec{A}_1 &= A,\\[1mm] \vec{A}_2 &= A\cos60^\circ + i\,A\sin60^\circ = \frac{A}{2} + i\frac{\sqrt{3}A}{2},\\[1mm] \vec{A}_3 &= A\cos120^\circ + i\,A\sin120^\circ = -\frac{A}{2} + i\frac{\sqrt{3}A}{2}. \end{aligned}$$

Summing:

$$\begin{aligned} \text{Real part} &= A + \frac{A}{2} - \frac{A}{2} = A,\\[1mm] \text{Imaginary part} &= \frac{\sqrt{3}A}{2}+\frac{\sqrt{3}A}{2}= \sqrt{3}A. \end{aligned}$$

Thus, the resultant phasor is:

$$\vec{R} = A + i\sqrt{3}A.$$

Step 2: Resultant amplitude and phase

Magnitude (Amplitude):

$$|\vec{R}|=\sqrt{A^2+( \sqrt{3}A)^2}=\sqrt{A^2+3A^2} =2A.$$

Phase:

$$\phi=\tan^{-1}\left(\frac{\sqrt{3}A}{A}\right)=60^\circ.$$

Step 3: Energy considerations

Energy in SHM ∝ (Amplitude)$^2$. Hence, the energy of the resulting motion:

$$E_R\propto (2A)^2=4A^2,$$

which is 4 times that of any single motion $(A^2)$.

Step 4: Nature of the resultant motion

Since the summation gives a single cosine function with amplitude $2A$ and phase $60^\circ$, the motion is indeed simple harmonic.