B M Sharma Solutions for Chapter: Dimensions and Measurement, Exercise 54: Archives

A person measures the depth of a well by measuring the time interval between dropping a stone and receiving the sound of impact with the bottom of the well. The error in his measurement of time is $\mathrm{\delta }T=0.01$ second and he measures the depth of the well to be $L=20 \mathrm{meters}$. Take the acceleration due to gravity $g=10{ \mathrm{m} \mathrm{s}}^{-2}$ and the velocity of sound is $300{ \mathrm{m} \mathrm{s}}^{-1}$. Then the fractional error in the measurement $\frac{\mathrm{\delta }L}{L}$ is closest to

A student uses a simple pendulum of exactly $1 \mathrm{m}$ length to determine $g$, the acceleration due to gravity. He uses a stop watch with the least count of $1 \mathrm{s}$ for this and records $40 \mathrm{s}$ for $20$ oscillations. For this observation, which of the following statement(s) is(are) true?

Planck's constant $h$, speed of light $c$ and gravitational constant $G$ are used to form a unit of length $L$ and a unit of mass $M$. Then the correct options(s) is (are)

In terms of potential difference $V$, electric current $I$, permittivity ${\epsilon }_0$, permeability ${\mu }_0$ and speed of light $c$, the dimensionally correct equations(s) is (are)

A length scale $\left(l\right)$ depends on the permittivity $\left(\epsilon \right)$ of a dielectric material, Boltzmann constant $k_{\mathrm{B}}$, the absolute temperature $T$, the number per unit volume $\left(n\right)$ of certain charged particles, and the charge $\left(q\right)$ carried by each of the particles. Which of the following expressions(s) for $l$ is(are) dimensionally correct?

In an experiment to determine the acceleration due to gravity $g$, the formula used for the time period of a period of a periodic motion is, $T=2\pi \sqrt{\frac{7\left(R-r\right)}{5g}}$. The values of $R$ and $r$ are measured to be $\left(60\pm 1\right) \mathrm{mm}$ and $\left(10\pm 1\right) \mathrm{mm}$, respectively. In five successive measurements, the time period is found to be $0.52 \mathrm{s}\text{,} 0.56 \mathrm{s}\text{,} 0.57 \mathrm{s}\text{,} 0.54 \mathrm{s}$ and $0.59 \mathrm{s}$. The least count of the watch used for the measurement of time period is $0.01 \mathrm{s}$. Which of the following statement(s) is (are) true?

To find the distance $d$ over which a signal can be seen clearly in foggy conditions, a railways engineer uses dimensional analysis and assumes that the distance depends on the mass density $\rho$ of the fog, intensity $\left(\text{power}/\text{area}\right)$ $S$ of the light from the signal and its frequency $f$. The engineer finds that $d$ is proportional to $S^{\frac{1}{\mathrm{n}}}$. The values of $n$ is,

The energy of a system as a function of time $t$ is given as $E\left(t\right)=A^2\exp \left(-\alpha t\right)$, where $\alpha =0.2{ \mathrm{s}}^{-1}$. The measurement of $A$ has an error of $1.25\%$. If the error in the measurement of time is $1.50\%,$ the percentage error in the value of $E\left(t\right)$ at $t=5 \mathrm{s}$ is