Units, Dimensions and Errors

Two thin wires, Wire-1 of diameter $0.650$ mm and Wire-2 of unknown diameter $d$ are given. To obtain the value of $d$, the diameters of the two wires are measured with a screw gauge. The screw gauge has a pitch of $0.5$ mm and there are $100$ divisions on the circular scale (CS). The smallest division on the linear scale (LS) is $0.5$ mm. The table shows the readings of LS and CS for the measurements. The value of $d$ (in µm) is:

	LS (mm)	CS
Wire-1	0.5	42
Wire-2	1.5	95

In a single slit diffraction experiment, a slit of width $(0.016\pm0.002)$ mm is used to measure the wavelength of a monochromatic light source. In the diffraction pattern, the angular distance between the central maximum and first minimum is measured to be $(2^\circ\pm40')$. The value of the fractional error in the measurement of wavelength is:
[Given: $\sin(2^\circ)=0.035$]

In a new system of units, the units of mass, length, time and current are $5$ kg, $5$ m, $5$ s and $5$ A, respectively. If $\mu_0$ and $\epsilon_0$ are the permeability and permittivity of free space, respectively, then in this new system of units, the magnitude of one SI unit of $\sqrt{\mu_0/\epsilon_0}$, is:

The dimensional formula of $\dfrac12\epsilon_0E^2$ ($\epsilon_0$ = permittivity of vacuum and $E$ = electric field) is $M^aL^bT^c$. The value of $2a-b+c$ = ________.

The diameter of a wire measured by a screw gauge of least count 0.001 cm is 0.08 cm. The length measured by a scale of least count 0.1 cm is 150 cm. When a weight of 100 N is applied to the wire, the extension in length is 0.5 cm, measured by a micrometer of least count 0.001 cm. The error in the measured Young's modulus is $\alpha\times10^9$ N/m². The value of $\alpha$ is ________.
(Ignore the contribution of the load to Young's modulus error calculation)

Dimensions of universal gravitational constant ($G$) in terms of Planck's constant ($h$), distance ($L$), mass ($M$) and time ($T$) are ________.

In a screw gauge the zero of main scale reference line coincides with the fifth division of the circular scale when two studs are in contact. There are 100 divisions in circular scale and pitch of screw gauge is 0.1 mm. When diameter of a sphere is measured, the reading of main scale is 5 mm and 50th division of circular scale coincides with the reference line of main scale. The diameter of sphere is ________ mm.

In a screw gauge when the circular scale is given five complete rotations it moves linearly by 2.5 mm. If the circular scale has 100 divisions, the least count of screw gauge is ________ mm.

Match the LIST-I with LIST-II

Choose the correct answer from the options given below:

In a Vernier calipers, when both jaws touch each other, zero of the Vernier scale is shifted to the right of zero of the main scale and 7th Vernier division coincides with a main scale reading. If the value of 1 main scale division is 1 mm and there are 10 Vernier scale divisions, then the Vernier caliper has

$L$, $C$ and $R$ represents physical quantities inductance, capacitance and resistance respectively. The dimensional formula $ML^2T^{-4}A^{-2}$ corresponds to ________.

Match List - I with List - II.

where $h$ (Planck's constant), $G$ (gravitational constant) and $c$ (speed of light in vacuum) as fundamental units.

Choose the correct answer from the options given below :

In an experiment to determine the resistance of a given wire using Ohm's law, the voltmeter and ammeter readings are noted as 10 V and 5 A, respectively. The least counts of voltmeter and ammeter are 500 mV and 200 mA, respectively. The estimated error in the resistance measurement is ________ Ω

The density $\rho$ of a uniform cylinder is determined by measuring its mass $m$, length $l$ and diameter $d$. The measured values of $m$, $l$ and $d$ are $97.42\pm0.02$ g, $8.35\pm0.05$ mm and $20.20\pm0.02$ mm, respectively. Calculated percentage fractional error in $\rho$ is ________.

The potential energy of a particle changes with distance $x$ from a fixed origin as $V=\dfrac{A\sqrt x}{x+B}$, where $A$ and $B$ are constant with appropriate dimensions. The dimensions of $AB$ are ________.

The percentage error in the calculated volume of a sphere, if there is 2% error in its diameter measurement, is ________.

Match List - I with List - II.

Choose the correct answer from the options given below :

A new unit ($\alpha$) of length is chosen such that it is equal to the speed of light in vacuum. What is the distance between Venus and Earth in terms of $\alpha$ units if light takes 6 min. 40 s to cover this distance ?

Consider the equation $H=\dfrac{x^p\epsilon^qE^r}{t^s}$

Where $H$ = magnetic field; $E$ = electric field, $\epsilon$ = permittivity, $x$ = distance, $t$ = time

The values of $p$, $q$, $r$ and $s$ respectively are :

Figure 1 shows the configuration of main scale and Vernier scale before measurement. Fig. 2 shows the configuration corresponding to the measurement of diameter $D$ of a tube. The measured value of $D$ is: