$A$ bomb explodes at time $t=0$ in a uniform,isotropic medium of density $\rho$ and releases energy $E$,generating a spherical blast wave. The radius $R$ of this blast wave varies with time $t$ as

Let us consider a system of units in which mass and angular momentum are dimensionless. If length has dimension of $L$,which of the following statement$(s)$ is/are correct?
$(1)$ The dimension of force is $L^{-3}$
$(2)$ The dimension of energy is $L^{-2}$
$(3)$ The dimension of power is $L^{-5}$
$(4)$ The dimension of linear momentum is $L^{-1}$

Consider the equation $\frac{1}{2} m v^{2} = m g h$,where $m$ is the mass of the body,$v$ is the velocity,$g$ is the acceleration due to gravity,and $h$ is the height. Is this equation dimensionally correct?

$A$ force $F$ is given by $F = at + bt^2$,where $t$ is time. What are the dimensions of $a$ and $b$?

The expressions below give current $I$ through an electronic component as a function of applied potential $V$. $I_0$ and $V_0$ are constants having dimensions of current and potential respectively. Which of the following are dimensionally incorrect?
$(A)$ $I=I_0\left(e^{\frac{2 V}{V_0}}+1\right)$
$(B)$ $I=I_0\left(e^{\frac{V}{2 V_0}}-1\right)$
$(C)$ $I=I_0 V_0\left(e^{\frac{V}{V_0}}-1\right)$
$(D)$ $I=I_0\left(\frac{V}{V_0}\right)\left(e^{\frac{V}{V_0}}-1\right)$

The speed of light $(c)$,gravitational constant $(G)$,and Planck's constant $(h)$ are taken as fundamental units in a system. The dimensions of time in this new system should be