Units, Dimensions and Errors

Match List - I (Event) with List - II (Order of the time interval for happening of the event) and select the correct option from the options given below the lists.

Using the expression $2d\sin\theta=\lambda$, one calculates the values of $d$ by measuring the corresponding angles $\theta$ in the range 0 to 90°. The wavelength $\lambda$ is exactly known and the error in $\theta$ is constant for all values of $\theta$. As $\theta$ increases from 0°,

Match List I with List II and select the correct answer using the codes given below the lists:

In the determination of Young's modulus $\left(Y=\dfrac{4MLg}{\pi ld^2}\right)$ by using Searle's method, a wire of length $L=2$ m and diameter $d=0.5$ mm is used. For a load $M=2.5$ kg, an extension $l=0.25$ mm in the length of the wire is observed. Quantities $d$ and $l$ are measured using a screw gauge and a micrometer, respectively. They have the same pitch of 0.5 mm. The number of divisions on their circular scale is 100. The contributions to the maximum probable error of the $Y$ measurement

A meter bridge is set-up as shown, to determine an unknown resistance 'X' using a standard 10 ohm resistor. The galvanometer shows null point when tapping-key is at 52 cm mark. The end-corrections are 1 cm and 2 cm respectively for the ends A and B. The determined value of 'X' is

Taking the electronic charge as '$e$' and the permittivity as '$\varepsilon_0$', use dimensional analysis to determine the correct expression for $\omega_p$.

Estimate the wavelength at which plasma reflection will occur for a metal having the density of electrons $N\approx4\times10^{27}\ \mathrm{m^{-3}}$. Take $\varepsilon_0\approx10^{-11}$ and $m\approx10^{-30}$, where these quantities are in proper SI units.

The density of a solid ball is to be determined in an experiment. The diameter of the ball is measured with a screw gauge, whose pitch is 0.5 mm and there are 50 divisions on the circular scale. The reading on the main scale is 2.5 mm and that on the circular scale is 20 divisions. If the measured mass of the ball has a relative error of 2%, the relative percentage error in the density is

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

For periodic motion of small amplitude A, the time period T of this particle is proportional to

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 cm. Least count for time = 0.1 s.

Student	Length of the pendulum (cm)	Number of oscillations ($n$)	Total time for ($n$) oscillations (s)	Time period (s)
I	64.0	8	128.0	16.0
II	64.0	4	64.0	16.0
III	20.0	4	36.0	9.0

If $E_I$, $E_{II}$ and $E_{III}$ are the percentage errors in $g$, i.e., $\left(\dfrac{\Delta g}{g}\times100\right)$ for students I, II and III, respectively,

Some physical quantities are given in Column I and some possible SI units in which these quantities may be expressed are given in Column II. Match the physical quantities in Column I with the units in Column II and indicate your answer by darkening appropriate bubbles in the 4 × 4 matrix given in the ORS.

A student performs an experiment to determine the Young's modulus of a wire, exactly 2 m long, by Searle's method. In a particular reading, the student measures the extension in the length of the wire to be 0.8 mm with an uncertainty of $\pm0.05$ mm at a load of exactly 1.0 kg. The student also measures the diameter of the wire to be 0.4 mm with an uncertainty of $\pm0.01$ mm. Take $g=9.8\ \mathrm{m/s^2}$ (exact). The Young's modulus obtained from the reading is