Question

The displacement of a particle varies with time according to the relation $y = a \,sin\,\omega t + b\, cos \,\omega t$.

• A. The motion is oscillatory but not SHM
• B. The motion is SHM with amplitude $a + b$
• C. The motion is SHM with amplitude $a^2 + b^2$
• D. The motion is SHM with amplitude $\sqrt {a^2 + b^2}$

Answer 1

Answer: (D)

The motion is SHM with amplitude $\sqrt {a^2 + b^2}$

Answer 2

Given :$x = a\, sin\, \omega t + b \,cos\, \omega t \quad...\left(i\right)$ Let $a = A \,cos \,\phi\quad...\left(ii\right)$ and $b = A \,sin\, \phi \quad...\left(iii\right)$ Squaring and adding $(ii)$ and $(iii)$, we get $a^2 +b^2 = A^2 \, cos^2 \phi + A^2\,sin^2 \phi = A^2$ E $(i)$ can be written as $x = A \,cos \phi \,sin\omega t + A\, sin\,\phi\, cos\,\omega t$ $= A sin ( \omega t +\phi)$ It is equation of $SHM$ with amplitude $A = \sqrt {a^2 +b^2}$.