A particle of mass $m$ moves along $x-$ axis under a force which is given by:

$F\left(x\right)=-Ax\left[1-e^{-k{x}^2}\right]$

Where $x$ is the position; while $A, k$ are constants. If the amplitude of oscillation be $a,$ and the time period be $T$ then

Study the following statements and find which is/are true.

$\mathrm{I}.$ The motion of the pendulum with the effect of dissipative force is known as damped oscillation.

$\mathrm{II}.$ Swinging of a child in a swing (without external push) is an example of free oscillation.

Choose correct option.

What will be the displacement equation of the simple harmonic motion obtained by combining the motions?
$x_1=2\sin \omega t, x_2=4\sin \left(\omega t+\frac{\pi }{6}\right) \mathrm{a}\mathrm{n}\mathrm{d} x_3=6\sin \left(\omega t+\frac{\pi }{3}\right)$

(Note: $\varphi$ is an acute angle)

A body of mass $0.01 \mathrm{kg}$ executes simple harmonic motion about $x=0$ under the influence of a force shown below. The period of the SHM is approximately

In the figure shown, the time period and the amplitude, respectively, when mass $m$ is released from rest and when the spring is relaxed are (The inclined plane is smooth)

A mass $m$ is suspended by means of two coiled springs which have the same length in unstretched condition, as shown in figure. Their force constants are $k_1$ and $k_2$, respectively. When set into vertical vibrations, the period will be