Ncert Exercises & Solutions

The equation of motion of a particle is $x =a \text{cos} ( \alpha t )^{2}$ . The motion is:

1. periodic but not oscillatory

2. periodic and oscillatory

3. oscillatory but not periodic

4. neither periodic nor oscillatory

3. Hint: The displacement of the particle is the same after a time period in a periodic motion.

Step 1: Find the position of the particle after one time period.

As the given equation is,

x =

a cos {(α t)}^{2}

$a \cos {(α t)}^{2}$ is a cosine function. Hence, it is an oscilatory motion.

Now, putting t+T in place of t,

x(t+T)=a cos

${[α (t + T)]}^{2} [∵ x (t) = a \cos {(a t)}^{2}]$

$= a \cos [α^{2} t^{2} + α^{2} T^{2} + 2 α t T] \neq x (t)$

where, T is supposed as period of the function

$ω$ (t).

Hence, it is not periodic.