Ncert Exercises & Solutions

Show that for a particle executing SHM, the velocity and displacement have a phase difference of $π$ $π$ /2.

Hint: Use the equation of SHM.

Step 1: Find the equation of displacement.

Let us assume, the displacement function of SHM:

x= acos

ω t

$ω t$

where a=amplitude of motion

Step 2: Find the equation of velocity.

velocity, v

= \frac{d x}{d t}

$= \frac{d x}{d t}$

\frac{d x}{d t} = a (- \sin ω t) ω = - ω a \sin ω t

$\frac{d x}{d t} = a (- \sin ω t) ω = - ω a \sin ω t$

or v=

- ω a \sin ω t

$- ω a \sin ω t$

= ω a \cos (\frac{π}{2} + ω t) [∵ \sin ω t = - \cos (\frac{π}{2} + ω t)]

$= ω a \cos (\frac{π}{2} + ω t) [∵ \sin ω t = - \cos (\frac{π}{2} + ω t)]$

Step 3: Find the phase difference.

Now, the phase of displacement =

ω t

$ω t$

the phase of velocity

= \frac{π}{2} + ω t

$= \frac{π}{2} + ω t$

∴

$∴$ The difference in the phase of velocity to that of the phase of displacement

= \frac{π}{2} + ω t - ω t = \frac{π}{2}

$= \frac{π}{2} + ω t - ω t = \frac{π}{2}$