Question

The motion of a particle varies with time according to the relation y = a sin ω t + a cos ω t. Then, ______. 

Options

• the motion is oscillatory but not SHM 

• the motion is SHM with amplitude a 

• the motion is SHM with amplitude `sqrt(2"a")` 

• None of the above

Answer 1

The motion of a particle varies with time according to the relation y = a sin ω t + a cos ω t. Then, `bbunderline("the motion is SHM with amplitude" sqrt(2"a"))`.

Explanation:

We know,

By definition,

A motion is SHM, when

a ∝ − x

Where, a is restoring acceleration due to restoring force x us displacement from mean position.

Here Given,

y = a sin(ωt) + a cos(ωt)

v = `"dy"/"dt"` = a cos(ωt) − a sin(ωt)

a = `"dv"/"dt"=("d"^2"y")/"dt"^2` = −a sin(ωt) − a cos(ωt) = −y

a = −y

Hence, it is proved that y = a sin(ωt) + a cos(ωt) is a SHM.

Also, amplitude is defined as the maximum displacement from the mean position (here y = 0).

Maximum value of y = `sqrt("a"^2+"a"^2)=sqrt(2"a")`