Umakant Kondapure, Collin Fernandes, Nipun Bhatia, Vikram Bathula and, Ketki Deshpande Solutions for Chapter: Measurements, Exercise 3: Competitive Thinking

Students I, II and III perform an experiment for measuring the acceleration due to gravity ($g$) using a simple pendulum. They use different lengths of the pendulum and/or record time for different number of oscillations. The observations are shown in the table. Least count for length $=0.1\mathrm{cm}$. Least count for time $=0.1\mathrm{s}$.

$\begin{array}{|llccc|}\hline & \begin{array}{c}\text{ Length of the }\\ \text{ pendulum (cm)}\end{array}& \begin{array}{c}\text{ Number of }\\ \text{ oscillation (}n\text{) }\end{array}& \begin{array}{c}\text{ Total time }\\ \text{ for (}n\text{) }\\ \text{ oscillations }\end{array}& \begin{array}{c}\text{ Time }\\ \text{ period }\\ \text{ (s) }\end{array}\\ \text{ I }& 64.0& 8& 128.0& 16.0\\ \text{ II }& 64.0& 4& 64.0& 16.0\\ \text{ III }& 20.0& 4& 36.0& 9.0\\ \hline\end{array}$

If $E_1, E_{11}$ and $E_{111}$ are the percentage errors in $g$, i.e., $\left(\frac{\mathrm{\Delta }g}{\mathrm{g}}\times 100\right)$ for students $\mathrm{I}$, $\mathrm{II}$ and $\mathrm{III}$, respectively.
 

One femtometer is equivalent to

Which of the following units denotes the dimensions $\frac{{\mathrm{ML}}^2}{Q^2}$ where $Q$ denotes the electric charge? 
 

Which of the following physical quantities represent the dimensions of $\frac{b}{a}$ in the relation $P=\frac{x^2-b}{at},$ where, $P$ is power, $x$ is distance and $t$ time?

If force $F$, velocity $V$ and time $T$ are taken as fundamental units, then the dimensions of mass are
 

If energy $E$, velocity $V$ and time $T$ are chosen as the fundamental quantities, the dimensional formula of surface tension will be
 

If the velocity of the surface wave $\left(v\right)$ depends upon surface tension $\left(T\right)$, coefficient of viscosity $\left(\eta \right)$ and density $\left(\rho \right),$ then the expression for $v$ will be 

The ratio of the dimensions of Planck constant and that of moment of inertia has the dimensions of