$A$ particle executing $SHM$ of amplitude $a$ has a displacement of $-\frac{a}{2}$ at $t = \frac{T}{4}$ and a positive velocity. Find the initial phase of the particle.

$A$ particle executes simple harmonic motion according to the equation $x(t) = A \sin^2(\alpha t)$. If the time period of the $SHM$ is $0.2 \ s$, then the value of $\alpha$ (in units of $rad/s$) is (in $pi$)

The displacement of a particle in an oscillatory motion at a time $t$ is given by $x = 8 \sin \frac{\pi t}{4} \text{ cm}$. Calculate its displacement in the time interval $t = 0 \text{ s}$ to $t = 2 \text{ s}$. (in $\text{ cm}$)

Linear simple harmonic motion is a projection of uniform circular motion along any one diameter of a circle. Is there any distinction between these two?

Consider a one-dimensional potential $V(x)$ as shown in the figure below. $A$ classical particle of mass $m$ moves under its influence and has total energy $E$ as shown below. The motion is

The figures correspond to two circular motions. The radius of the circle,the period of revolution,the initial position,and the sense of revolution (i.e.,clockwise or anti-clockwise) are indicated on each figure. Obtain the corresponding simple harmonic motions of the $x$-projection of the radius vector of the revolving particle $P$,in each case.