Question

The motion of a particle varies with time as y=a(sinωt+cosωt), then

• A. the motion is oscillatory but not S.H.M.
• B. the motion is S.H.M. with amplitude '$a$'
• C. the motion is S.H.M. with amplitude $a\sqrt{2}$
• D. the motion is S.H.M. with amplitude
  2$a$

Answer 1

Answer: (C)

the motion is S.H.M. with amplitude $\sqrt{2}a$

Given,

Motion is varies as y=a(sinωt+cosωt)

We can write,

$y=asinwt+asin(90-wt)$

$Or,y=2asin\frac{(wt+90-wt)}{2}+cos\frac{(wt+90-wt)}{2}$

Or,

$y=2asin45cos(45-\frac{wt}{2})$

$y=\sqrt{2}acos(45-\frac{wt}{2})$

From the above equation, we can say that the particle is performing SHM and after comparing we can also say that the amplitude of the SHM is

$\sqrt{2}a$