Question

Two simple harmonic motions with the same frequency act on a particle at right angles i.e., along x and y axis. If the two amplitudes are equal and the phase difference is $ \pi $ /2, the resultant motion will be :

• A. a circle
• B. an ellipse with the major axis along y-axis
• C. an ellipse with the major axis along x-axis
• D. a straight line inclined at 45? to the x-axis

Answer 1

Answer: (A)

The two simple harmonic motions can be given by $ x=a\,\sin \omega t $ ?...(i) and $ y=a\sin \left( \omega t+\frac{\pi }{2} \right) $ $ y=a\,\cos \,\omega t $ ..?.(ii) On squaring and adding Eqs. (i) and (ii), we obtain $ {{x}^{2}}+{{y}^{2}}={{a}^{2}}({{\sin }^{2}}\,\omega t+{{\cos }^{2}}\omega t) $ or $ {{x}^{2}}+{{y}^{2}}={{a}^{2}} $ This is the equation of a circular motion with radius a. NOTE: Simple harmonic motion is of two types : 1. Linear simple harmonic motion 2. Angular simple harmonic motion