Units & Measurements

In a new system of units, the units of mass, length, time and current are $5 \mathrm{~kg}, 5 \mathrm{~m}, 5 \mathrm{~s}$ and 5 A , respectively. If $\mu_0$ and $\epsilon_0$ are the permeability and permittivity of free space, respectively, ...

In a single slit diffraction experiment, a slit of width $(0.016 \pm 0.002)$ 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 minimu...

A temperature difference can generate e.m.f. in some materials. Let S be the e.m.f. produced per unit temperature difference between the ends of a wire, σ the electrical conductivity and κ the thermal conductivity of the material of the wir...

Length, breadth and thickness of a strip having a uniform cross section are measured to be 10.5 cm, 0.05 mm, and 6.0 μm, respectively. Which of the following option(s) give(s) the volume of the strip in cm3 with correct significant figures:

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:

The dimensions of a cone are measured using a scale with a least count of $2 \mathrm{~mm}$. The diameter of the base and the height are both measured to be $20.0 \mathrm{~cm}$. The maximum percentage error in the determination of the volume...

A dimensionless quantity is constructed in terms of electronic charge $e$, permittivity of free space $\varepsilon_0$, Planck's constant $h$, and speed of light $c$. If the dimensionless quantity is written as $e^\alpha \varepsilon_0{ }^\be...

Young's modulus of elasticity $Y$ is expressed in terms of three derived quantities, namely, the gravitational constant $G$, Planck's constant $h$ and the speed of light $c$, as $Y=c^\alpha h^\beta G^\gamma$. Which of the following is the c...

In an experiment for determination of the focal length of a thin convex lens, the distance of the object from the lens is $10 \pm 0.1 \mathrm{~cm}$ and the distance of its real image from the lens is $20 \pm 0.2 \mathrm{~cm}$. The error in ...

Area of the cross-section of a wire is measured using a screw gauge. The pitch of the main scale is $0.5 \mathrm{~mm}$. The circular scale has 100 divisions and for one full rotation of the circular scale, the main scale shifts by two divi...

In a particular system of units, a physical quantity can be expressed in terms of the electric charge $e$, electron mass $m_{e}$, Planck's constant $h$, and Coulomb's constant $k=\frac{1}{4 \pi \epsilon_{0}}$, where $\epsilon_{0}$ is the pe...

A physical quantity \(\overrightarrow S\) is defined as \(\overrightarrow S = (\overrightarrow E \times \overrightarrow B )/{\mu _0}\), where \(\overrightarrow E\) is electric field, \(\overrightarrow B\) is magnetic field and \(\mu\)0...

The smallest division on the main scale of a Vernier calipers is 0.1 cm. Ten divisions of the Vernier scale correspond to nine divisions of the main scale. The figure below on the left shows the reading of this calipers with no gap between ...

Sometimes it is convenient to construct a system of units so that all quantities can be expressed in
terms of only one physical quantity. In one such system, dimensions of different quantities are given
in terms of a quantity X as follows: ...

An optical bench has 1.5 m long scale having four equal divisions in each cm. While measuring the focal length of a convex lens, the lens is kept at 75 cm mark of the scale and the object pin is kept at 45 cm mark. The image of the object p...

Let us consider a system of units in which mass and angular momentum are dimensionless. If length has dimension of L, which of the following statement(s) is/are correct?

A steel wire of diameter \(0.5\) \(mm\) and Young's modulus \(2 \times {10^{11}}\,\,N{m^{ - 2}}\) carries a load of mass \(M.\) The length of the wire with the load is \(1.0\) \(m.A\) vernier scale with \(10\) divisions is attached to the e...

If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the ...

If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the ...

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 \(\delta T = 0.01\) seconds and he measures...

In an experiment to determine the acceleration due to gravity g, the formula used for the time period of a
periodic motion is \(T = 2\pi \sqrt {{{7\left( {R - r} \right)} \over {5g}}}\). The values of R and r are measured to be (60 $$ \pm ...

There are two Vernier calipers both of which have 1 cm divided into 10 equal divisions on the main scale.
The Vernier scale of one of the calipers (C1) has 10 equal divisions that correspond to 9 main scale
divisions. The Vernier scale of t...

A length-scale (l) depends on the permittivity (\(\varepsilon\)) of a dielectric material, Boltzmann constant (kB), the absolute temperature (T), the number per unit volume (n) of certain charged particles, and the charge (q) carried by ea...

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

Consider a Vernier callipers in which each 1 cm on the main scale is divided into 8 equal divisions and a
screw gauge with 100 divisions on its circular scale. In the Vernier callipers, 5 divisions of the Vernier
scale coincide with 4 divis...

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 option(s) is(are)

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
(power/area) S of th...

During Searle's experiment, zero of the Vernier scale lies between 3.20 \(\times\) 10-2 m and 3.25 \(\times\) 10-2 m of the
main scale. The 20th division of the Vernier scale exactly coincides with one of the main scale divisions.
When ...

Match List I with List II and select the correct answer using the codes given below the lists:
List I
P. Boltzmann Constant
Q. Coefficient of viscosity
R. Plank Constant
S. Thermal conductivity
List II
1. [ML2T-1]
2. [ML-1T-1]
3. [MLT-3K-1]...

The diameter of a cylinder is measured using a Vernier callipers with no zero error. It is found that the zero
of the Vernier scale lies between 5.10 cm and 5.15 cm of the main scale. The Vernier scale has 50 divisions
equivalent to 2.45 cm...

In the determination of Young's modulus \(\left( {Y = {{4MLg} \over {\pi l{d^2}}}} \right)\)
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 o...

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 t...

A dense collection of equal number of electrons and positive ions is called neutral plasma. Certain solids containing
fixed positive ions surrounded by free electrons can be treated as neutral plasma. Let 'N' be the number density of
free e...

A vernier calipers has 1 mm marks on the main scale. It has 20 equal divisions on the Vernier scale which match with 16 main scale divisions. For this Vernier calipers, the least count 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 th...

Student I, II and III perform an experiment for measuring the acceleration due to gravity (g) using a
simple pendulum. They use different length of the pendulum and/or record time for different number
of oscillations. The observations area ...

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 $$\...

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.
Column I
(A) GMeMs ,
G $$ \t...

A student performs an experiment for determination of $g\left(=\frac{4 \pi^2 l}{\mathrm{~T}^2}\right), l=1 m$, and he commits an error of $\Delta l$. For T , he takes the time of $n$ oscillations with the stop watch of least count $\Delta \...

In a screw gauge, the zero of main scale coincides with the fifth division of circular scale in figure (i).The circular division of screw gauge is 50. It moves 0.5 mm on main scale in one rotation.The diameter of the ball in figure (ii) is

Which of the following set have different dimensions?

The side of a cube is measured by vernier
callipers (10 divisions of a vernier scale
coincide with 9 divisions of main scale,
where 1 division of main scale is 1 mm). The
main scale reads 10 mm and first division of
vernier scale coinc...

The side of a cube is measured by vernier callipers (10 divisions of a vernier scale coincide with
9 divisions of main scale, where 1 division of main scale is 1 mm). The main scale reads 10 mm and
first division of vernier scale coincides ...

Pressure depends on distance as, \(P = {\alpha \over \beta }\exp \left( { - {{\alpha z} \over {k\theta }}} \right)\), where \(\alpha\), \(\beta\) are constants, z in distance, k is Boltzman's constant and \(\theta\) is temperature. The ...

A wire of length \(l = 6 \pm 0.06\) cm and \(r = 0.5 \pm 0.005\) cm and mass \(m = 0.3 \pm 0.003\) gm. Maximum percentage error in density is

A screw gauge having 100 equal divisions and a pitch of length 1 mm is used to measure the diameter of a wire of length 5.6 cm. The main scale reading is 1mm and 47th circular division coincides
with the main scale. Find the curved surface ...

In Searle's experiment, which is used to find Young's Modulus of elasticity, the diameter of experimental wire is D = 0.05 cm ( measured by a scale of least count 0.001 cm ) and length is L = 110 cm ( measured by a scale of least count 0.1 ...

A cube has a side of length 1.2 ✕ 10-2 m. Calculate its volume.

If nth divisions of the main scale coincide with (n+1)th divisions
of vernier scale. Given one main scale division is equal to 'a' units. Find the
least count of the vernier.

A quantity X is given by \({\varepsilon _0}L{{\Delta V} \over {\Delta t}}\) where \({\varepsilon _0}\) is the permittivity of the free space, L is a length, \({\Delta V}\) is a potential difference and \({\Delta t}\) is a time interval. The...