With the usual notations,the following equation $S_t = u + \frac{1}{2}a(2t - 1)$ is

Consider the efficiency of Carnot's engine is given by $\eta = \frac{\alpha \beta}{\sin \theta} \log_{e} \frac{\beta x}{k T}$,where $\alpha$ and $\beta$ are constants. If $T$ is temperature,$k$ is Boltzmann constant,$\theta$ is angular displacement and $x$ has the dimensions of length. Then,choose the incorrect option.

If Surface tension $(S)$,Moment of Inertia $(I)$ and Planck's constant $(h)$ were to be taken as the fundamental units,the dimensional formula for linear momentum would be

The refractive index of the medium is $\mu = A + \frac{B}{\lambda^{2}}$,where $A$ and $B$ are constants and $\lambda$ is the wavelength of light. The dimensions of $B$ are the same as that of

In an experiment,four quantities $a, b, c,$ and $d$ are measured with percentage errors of $1\%, 2\%, 3\%,$ and $4\%$ respectively. The quantity $P$ is calculated as $P = \frac{a^3 b^2}{cd}$. The percentage error in $P$ is ........ $\%$

Consider a simple pendulum,having a bob attached to a string,that oscillates under the action of the force of gravity. Suppose that the period of oscillation of the simple pendulum depends on its length $(l)$,mass of the bob $(m)$,and acceleration due to gravity $(g)$. Derive the expression for its time period using the method of dimensions.