First slide

Periodic Motion

Question

The equation of motion of a particle is $x = a \cos {(αt)}^{2}$ . The motion is

Moderate

A

periodic but not oscillatory
B

periodic and oscillatory
C

oscillatory but not periodic
D

Neither periodic nor oscillatory

Solution

As the given equation is
$x = a \cos {(αt)}^{2}$
is a cosine function. Hence, it is an oscillatory motion.
Now, putting (t + T) in place of t,
$x (t + T) = a \cos [α (t + T)]^{2}$ $[∵ x (t) = a \cos (αt)^{2}]$
$= a \cos [{αt}^{2} + {αT}^{2} + 2 αtT] \neq x (t)$
where, T is supposed as period of the function $ω$ (t).
Hence, it is not periodic.

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