If $E$ and $E_0$ denote energies at time $t$ and $t_0$ respectively, and $L$ and $L_0$ denote distances from some point at $t$ and $t_0$ respectively, then which of the following equations can be declared to be incorrect on dimensional grounds?
$(A) E = \frac{2 E_0 L}{L_0}$
$(B) E = E_0 e^{-\frac{2 L}{L_0}}$
$(C) E = 2 L e^{-\frac{L}{E_0}}$
$(D) E = 2 \left( \frac{E_0}{L_0} \right) e^{-\frac{L}{L_0}}$

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

Force $F$ is given in terms of time $t$ and distance $x$ by $F = a \sin(ct) + b \cos(dx)$. Then the dimension of $a/b$ is:

$A$ flywheel can rotate in order to store kinetic energy. The flywheel is a uniform disk made of a material with a density $\rho$ and tensile strength $\sigma$ (measured in Pascals),a radius $r$,and a thickness $h$. The flywheel is rotating at the maximum possible angular velocity so that it does not break. Which of the following expressions correctly gives the maximum kinetic energy per kilogram that can be stored in the flywheel? Assume that $\alpha$ is a dimensionless constant.

$A$ physical quantity $x$ is represented by the formula $x = M^a L^b T^c$. If $c \neq 0$,then:

Check that the ratio $ke^{2} / G m_{e} m_{p}$ is dimensionless. Look up a Table of Physical Constants and determine the value of this ratio. What does the ratio signify?