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Q.
The displacement of a particle varies with time according to the relation $y = a \,sin\,\omega t + b\, cos \,\omega t$.
Oscillations
Solution:
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
$a^2 +b^2 = A^2 \, cos^2 \phi + A^2\,sin^2 \phi = A^2$
Eq. $(i)$ 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
$A = \sqrt {a^2 +b^2}$.