Let us consider a periodic SHM is represented as $x=a \sin \omega t$
Assume mass of the body is $m$ is executing the SHM.
(a) Total mechanical energy of the body at any time is= K.E.max.
$$
\text { T.E. }=\frac{1}{2} m \omega^2 a^2
$$
Kinetic energy at any instant $t$ is
$$
\begin{aligned}
\text { K.E. } &=\frac{1}{2} m v^2=\frac{1}{2} m\left[\frac{d x}{d t}\right]^2 \quad\left[\because v=\frac{d x}{d t}\right] \\
&=\frac{1}{2} m \omega^2 a^2 \cos ^2 \omega t \quad[\because x=a \sin \omega t]
\end{aligned}
$$
On average total energy is K.E. (max.) $^{-1}$
$$
\text { K.E. }{ }_{\cdot \max .}=\frac{1}{2} m \omega^2 a^2=E
$$
(b) If amplitude is $a$ and angular frequency is $\omega$, then maximum velocity of particle will be equal to $(a \omega)$ and it varies according to sine law. Hence, rms value of velocity of the particle over $a$ complete cycle will be $\left(\frac{1}{\sqrt{2}} \vec{a} \cdot \vec{\omega}\right)$
Average $\left(\mathrm{KE}_{\mathrm{av}}\right)=\frac{1}{2} m v_{\mathrm{rms}}^2$
$$
=\frac{1}{2} m\left(\frac{1}{\sqrt{2}} \vec{a} \cdot \vec{w}\right)^2
$$
K.E. at any instant $t$ is
$$
\begin{aligned}
&\text { K.E.Av. }=\frac{1}{2} m \omega^2 a^2 \cos ^2 \omega t \\
&\begin{aligned}
\left(\mathrm{KE}_{\text {av }}\right) &=\frac{1}{2} m \omega^2 a^2\left[\left(\cos ^2 \omega t\right)_{a v}\right] \text { for a cycle } \\
\quad=\frac{1}{2} m \omega^2 a^2\left[\frac{0+1}{2}\right]
\end{aligned} \\
&\therefore \text { Average K.E. }=\frac{1}{4} m \omega^2 a^2=\frac{\mathrm{KE}_{\text {max. }}}{2}
\end{aligned}
$$
[from Eq. (ii)]
(c) Velocity $=v=\frac{d x}{d t}=a \omega \cos \omega t$
(For a complete cycle the mean velocity is)
$$
\begin{aligned}
v_{\text {mean }} &=\frac{v_{\max }+v_{\min .}}{2} \\
&=\frac{a \omega+(-a \omega)}{2}=0 \\
v_{\text {mean }} &=0 \quad\left(\because v_{\text {max }} \neq v_{\text {mean }}\right)
\end{aligned}
$$
(d)
$$
\begin{aligned}
&v_{\mathrm{mss}}=\sqrt{\frac{v_1^2+v_2^2}{2}}=\sqrt{\frac{0+a^2 \omega^2}{2}}=\frac{a \omega}{\sqrt{2}} \\
&v_{\mathrm{ms}}=\frac{v_{\max }^{\sqrt{2}}}{\sqrt{2}}
\end{aligned}
$$