Which of the following relationships between the acceleration $a$ and the displacement $x$ of a particle involve simple harmonic motion?
$(a)\; a=0.7 x$
$(b)\; a=-200 x^{2}$
$(c)\; a=-10 x$
$(d)\; a=100 x^{3}$

The function $x = A \sin^2 \omega t + B \cos^2 \omega t + C \sin \omega t \cos \omega t$ does not represent $SHM$ for which of the following conditions?

$A$ particle of mass $0.1 \ kg$ is executing simple harmonic motion of amplitude $0.1 \ m$. When the particle passes through the mean position,its kinetic energy is $8 \times 10^{-3} \ J$. If the initial phase is $45^{\circ}$,the equation of its motion is (Assume $x(t)$ as the position of the particle at time $t$)

$A$ particle executing simple harmonic motion of amplitude $5\, cm$ has a maximum speed of $31.4\, cm/s$. The frequency of its oscillation is ..... $Hz$.

The equation of motion of a particle is $\frac{d^2y}{dt^2} + Ky = 0$,where $K$ is a positive constant. The time period of the motion is given by

The motion of a particle executing $S.H.M.$ is given by $x = 0.01 \sin 100\pi (t + 0.05)$,where $x$ is in metres and $t$ is in seconds. The time period is ..... $sec$.