Let force $F = A \sin(Ct) + B \cos(Dx)$,where $x$ and $t$ are displacement and time respectively. The dimensions of $\frac{C}{D}$ are the same as the dimensions of

If the gravitational constant $(G)$,Planck's constant $(h)$,and the velocity of light $(c)$ are chosen as fundamental units,the dimension of the radius of gyration is:

In the relation $P = \frac{\alpha}{\beta} e^{-\frac{\alpha Z}{k\theta}}$,$P$ is pressure,$Z$ is distance,$k$ is Boltzmann constant,and $\theta$ is temperature. The dimensional formula of $\beta$ is:

An expression of energy density is given by $u = \frac{\alpha}{\beta} \sin \left(\frac{\alpha x}{k t}\right)$,where $\alpha, \beta$ are constants,$x$ is displacement,$k$ is Boltzmann constant,and $t$ is the temperature. The dimensions of $\beta$ will be.

In the relation $y = a \cos (\omega t - kx)$,the dimensional formula for $k$ is

$A$ calorie is a unit of heat or energy and it equals about $4.2 \; J$ where $1 \; J = 1 \; kg \; m^2 \; s^{-2}$. Suppose we employ a system of units in which the unit of mass equals $\alpha \; kg$, the unit of length equals $\beta \; m$, and the unit of time equals $\gamma \; s$. Show that a calorie has a magnitude $4.2 \; \alpha^{-1} \beta^{-2} \gamma^2$ in terms of the new units.