Chapter 1

The equation of alternating current is \(I=I_{0} e^{-t / C R}\) where \(t\) is time, \(C\) is capacitance and \(R\) is resistance of coil, then the dimensions of \(C R\) is (a) \(\left[\mathrm{MLT}^{-1}\right]\) (b) \(\left[\mathrm{M}^{0} \mathrm{LT}\right]\) (c) \(\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}\right]\) (d) None of these

Dimensions of which base quantity corresponds to that of \(\sqrt{\frac{G h}{c^{3}}}=\) ? (a) Time (b) Length (c) Mass (d) Temperature

Which of the following pairs has same dimensions? (a) Current density and charge density (b) Angular momentum and momentum (c) Spring constant and surface energy (d) Force and torque

The dimensions of distance travelled in \(n\)th second are (a) \(\left[\mathrm{M}^{0} \mathrm{LT}\right]\) (b) \(\left[\mathrm{M}^{\mathrm{O}} \mathrm{L}^{0} \mathrm{~T}^{0}\right]\) (c) \(\left[\mathrm{M}^{0} \mathrm{LT}^{-1}\right]\) (d) \(\left[\mathrm{M}^{0} \mathrm{LT}^{0}\right]\)

How many wavelengths of \(\mathrm{Kr}^{86}\) are there in one metre? (a) \(1553164.13\) (b) \(1650763.73\) (c) \(652189.63\) (d) \(2348123.73\)

The dimensions of universal gravitational constant are (a) \(\left[\mathrm{ML}^{-3} \mathrm{~T}^{2}\right]\) (b) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-3}\right]\) (c) \(\left[\mathrm{M}^{-1} \mathrm{~L}^{3} \mathrm{~T}^{-2}\right]\) (d) \(\left[\mathrm{M}^{2} \mathrm{~L}^{2} \mathrm{~T}^{-2}\right]\)

Taking frequency \(f\), velocity \(v\) and density \(\rho\) to be the fundamental quantities, then the dimensional formula for momentum will be (a) \(\left[\rho v^{4} f^{-1}\right]\) (b) \(\left[\rho v^{-1} f^{-1}\right]\) (c) \(\left[\rho v f^{2}\right]\) (d) \(\left[\rho^{2} v^{2} f^{2}\right]\)

Coefficient of thermal conductivity has the dimensions (a) \(\left[\mathrm{ML}^{-1} \mathrm{~T}^{3} \mathrm{~K}^{3}\right]\) (b) \(\left[\mathrm{ML}^{-1} \mathrm{~T}^{-3} \mathrm{~K}^{-1}\right]\) (c) \(\left[\mathrm{MLT}^{-3} \mathrm{~K}^{-1}\right]\) (d) \(\left[\mathrm{MLT}^{-3} \mathrm{~K}\right]\)

The dimensions of electrical conductivity are (a) \(\left[\mathrm{ML}^{3} \mathrm{~T}^{-3} \mathrm{~A}^{-2}\right]\) (b) \(\left[\mathrm{M}^{-1} \mathrm{~L}^{-3} \mathrm{~T}^{3} \mathrm{~A}^{2}\right]\) (c) \(\left[\mathrm{M}^{-1} \mathrm{~L}^{-3} \mathrm{~T}^{3} \mathrm{~A}^{2}\right]\) (d) \(\left[\mathrm{M}^{-1} \mathrm{~L}^{-3} \mathrm{~T}^{-3} \mathrm{~A}^{2}\right]\)

Farad is not equivalent to (a) \(\frac{q}{v}\) (b) \(q V^{2}\) (c) \(\frac{q^{2}}{J}\) (d) \(\frac{J}{v^{2}}\) \((q=\) coulomb, \(V=\) volt and \(J=\) joule \()\)

The dimensions of pole strength are (a) \(\left[\mathrm{M}^{\circ} \mathrm{LT}^{0} \mathrm{~A}\right]\) (b) \(\left[\mathrm{M}^{0} \mathrm{LTA}\right]\) (c) \(\left[\mathrm{M}^{\circ} \mathrm{L}^{-1} \mathrm{TA}^{-1}\right]\) (d) \(\left[\mathrm{M}^{0} \mathrm{~L}^{-1} \mathrm{~T}^{0} \mathrm{~A}^{-1}\right]\)

In the equation \(y=a \sin (\omega t+k x)\), the dimensional formula of \(\omega\) is (a) \(\left[\mathrm{M}^{\mathrm{O}} \mathrm{L}^{\mathrm{O}} \mathrm{T}^{-1}\right]\) (b) \(\left[\mathrm{M}^{0} \mathrm{LT}^{-1}\right]\) (c) \(\left[\mathrm{ML}^{\circ} \mathrm{T}^{\circ}\right]\) (d) \(\left[\mathrm{M}^{0} \mathrm{~L}^{-1} T^{0}\right]\)

Column I gives three physical quantities. Select the appropriate units for the choice given in Column II. Some of physical quantities may have more than one choice correct. Column I Column II A. Capacitance (p) Ohm-second B. Inductance (q) Coulmb \(^{2}\)-joule \(^{-1}\) C. Magnetic induction (r) Coulomb (volt) \(^{-1}\) (s) Newton (Ampere meter) \(^{-1}\) (t) Volt second (Ampere) \(^{-1}\) (a) \(\mathrm{q} \quad \mathrm{p} \quad \mathrm{s}\) (b) \(\mathrm{q} \quad \mathrm{r} \quad \mathrm{p}\) (c) \(\mathrm{r} \quad \mathrm{s} \quad \mathrm{p}\) (d) \(\mathrm{r}\) t \(\mathrm{q}\)

A new unit of length is chosen such that the speed of light in vacuum is unity. Then the distance between the sun and the earth in terms of the new unit, if light takes 8 min and \(20 \mathrm{~s}\) to cover this distance? [NCERT] (a) 300 new unit of length (b) 500 new unit of length (c) 600 new unit of length (d) None of these

Match the physical quantities given in column I with dimension expressed in terms of mass \((m)\), length \((L)\), time ( \(T\) ) and change \((Q)\) given in column II. Column I \(\quad\) Column II (A) Angular momentum (p) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\) (B) Torque (q) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-1}\right.\) ] (C) Inductance (r) \(\left[\mathrm{M}^{-1} \mathrm{~L}^{-2} \mathrm{~T}^{2} \mathrm{Q}^{2}\right]\) (D) Latent heat (s) \(\left[\mathrm{ML}^{2} \mathrm{Q}^{-2}\right]\) (E) Capacitance (t) \(\left[\mathrm{ML}^{3} \mathrm{~T}^{-1} \mathrm{Q}^{-2}\right]\) (F) Resistivity (u) \(\left[\mathrm{L}^{2} \mathrm{~T}^{-2}\right]\) \(\begin{array}{llllll}\mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{D} & \mathrm{E} & \mathrm{F}\end{array}\) (a) q s p t r u (b) q \(\quad \mathrm{p} \quad \mathrm{s} \quad \mathrm{u} \quad \mathrm{r} \quad \mathrm{t}\) (c) p \(\begin{array}{llllll}\text { (c) } & \text { u } & \text { r } & \text { t } & q\end{array}\) (d) \(\mathrm{s} \quad \mathrm{u}\) \(r\) \(\mathrm{t} \quad \mathrm{q} \quad \mathrm{p}\)

Assertion Impulse has the dimensions of force. Reason Impulse \(=\) force \(\times\) time.

\(\left[\mathrm{ML}^{-2} \mathrm{~T}^{-2}\right]\) represents dimensional formula of which of the following physical quantities? (a) Energy (b) Pressure (c) Torque (d) Pressure gradient

The length, breadth and thickness of a rectangular sheet of metal are \(4.234 \mathrm{~m}, 1.005 \mathrm{~m}\) and \(2.01 \mathrm{~cm}\) respectively. The area and volume of the sheet to correct significant figures are [NCERT] (a) \(8.72 \mathrm{~m}^{2}\) and \(0.0855 \mathrm{~m}^{3}\) (b) \(8.7 \mathrm{~m}^{2}\) and \(0.085 \mathrm{~m}^{3}\) (c) \(0.87 \mathrm{~m}^{2}\) and \(0.855 \mathrm{~m}^{3}\) (d) \(0.087 \mathrm{~m}^{2}\) and \(0.0855 \mathrm{~m}^{3}\)

The dimensions of emf in MKS is (a) \(\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2} \mathrm{Q}^{-2}\right]\) (b) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-2} \mathrm{Q}^{-2}\right]\) (c) \(\left[\mathrm{MLT}^{-2} \mathrm{Q}^{-1}\right]\) (d) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-2} \mathrm{Q}^{-1}\right]\)

Assertion Pressure has the dimensions of energy density. Reason Energy density \(=\frac{\text { energy }}{\text { volume }}=\frac{\left[M L^{2} T^{-2}\right]}{\left[L^{3}\right]}\) \(\left[\mathrm{ML}^{-1} \mathbf{T}^{-2}\right]=\) pressure

The physical quantity which has the dimensional formula \(\left[\mathrm{M}^{1} \mathrm{~T}^{-3}\right]\) is (a) surface tension (b) density (c) solar constant (d) compressibility

Assertion The unit used for measuring nuclear cross-section is barn'. Reason 1 barn \(=10^{-14} \mathrm{~m}^{2}\)

Force constant has same dimensions as (a) coefficient of viscosity (b) surface tension (c) frequency (d) impulse

In an experiment, the angles are required to be measured using an instrument. 29 divisions of the main scale coincide with 30 divisions of the vernier scale. If the smallest division of the main scale is half a degree \(\left(=0.5^{\circ}\right)\), then the least count of the instrument is (a) half minute (b) one degree (c) half degree (d) one minute

The dimensional formula of the ratio of angular to linear momentum is (a) \(\left[\mathrm{M}^{0} \mathrm{LT}^{0}\right]\) (b) [MLT] (c) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-1}\right]\) (d) \(\left[\mathrm{M}^{-1} \mathrm{~L}^{-1} \mathrm{~T}^{-1}\right]\)

If \(3.8 \times 10^{-6}\) is added to \(4.2 \times 10^{-5}\) giving due regard to significant figures, then the result will be [UP SEE 2009] (a) \(4.08 \times 10^{-5}\) (b) \(4.6 \times 10^{-5}\) (c) \(4.5 \times 10^{-5}\) (d) None of these

The maximum static friction on a body is \(F=\mu N\). Here, \(N=\) normal reaction force on the body \(\mu=\) coefficient of static friction. The dimensions of \(\mu\) are (a) \(\left[\mathrm{MLT}^{-2}\right]\) (b) \(\left[\mathrm{M}^{0} \mathrm{~L}^{\circ} \mathrm{T}^{0} \theta^{-1}\right]\) [c) dimensionless (d) None of these

One mole of an ideal gas at standard temperature and pressure occupies \(22.4 \mathrm{~L}\) (molar volume). The ratio of molar volume to the atomic volume of a mole of hydrogen? (Take the size of hydrogen molecule to be about \(1 \mathrm{~A}\) ) (a) \(9.1 \times 10^{4}\) (b) \(6 \times 10^{4}\) (c) \(7.1 \times 10^{4}\) (d) \(8.1 \times 10^{5}\)

Resistance of a given wire is obtained by measuring the current flowing in it and the voltage difference applied across it. If the percentage errors in the measurement of the current and the voltage difference are \(3 \%\) each, then error in the value of \mathrm{\\{} r e s i s t a n c e ~ o f ~ t h e ~ w i r e ~ i s ~ (a) \(6 \%\) (b) zero (c) \(1 \%\) (d) \(3 \%\)

If \(I\) is the moment of inertia and \(\omega\) the angular velocity, what is the dimensional formula of rotational kinetic energy (a) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-1}\right]\) (b) \(\left[\mathrm{M}^{2} \mathrm{~L}^{-1} \mathrm{~T}^{-2}\right]\) (c) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\) (d) \(\left[\mathrm{M}^{2} \mathrm{~L}^{-1} \mathrm{~T}^{-2}\right]\)

The respective number of significant figures for the number \(23.023,0.0003\) and \(21 \times 10^{-3}\) are [AIEEE 2010] (a) \(5,1,2\) (b) \(5,1,5\) (c) \(5,5,2\) (d) \(4,4,2\)

A gas bubble from an explosion under water oscillates with a time period \(T\), depends upon static pressure \(p\), density of water \(\rho\) and the total energy of explosion \(E\). The expression for the time period \(T\). (where, \(k\) is a dimensionless constant) is (a) \(T=k p^{-5 / 6} \rho^{1 / 2} E^{1 \phi}\) (b) \(T=k p^{-4 / 7} \rho^{t / 2} E^{1 / s}\) (c) \(T=k p^{-5 / 6} \rho^{1 / 2} E^{\prime / 2}\) (d) \(T=k p^{-4 \hbar} \rho^{1 / 2} E^{1 / 2}\)

Two full turns of the circular scale of a screw gauge cover a distance of \(1 \mathrm{~mm}\) on its main scale. The total number of divisions on the circular scale is 50 . Further, it is found that the screw gauge has a zero error of \(-0.03 \mathrm{~mm}\). While measuring the diameter of a thin wire, a student notes the main scale reading of \(3 \mathrm{~mm}\) and the number of circular scale divisions in line with the main scale as 35. The diameter of the wire is (a) \(3.32 \mathrm{~mm}\) (b) \(3.37 \mathrm{~mm}\) (c) \(3.67 \mathrm{~mm}\) (d) \(3.38 \mathrm{~mm}\)

Solar constant is defined as energy received by earth per \(\mathrm{cm}^{2}\) per minute. The dimensions of solar constant are (b) \(\left[\mathrm{M}^{2} \mathrm{~L}^{0} \mathrm{~T}^{-1}\right]\) (a) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-3}\right]\) (c) \(\left[\mathrm{ML}^{0} \mathrm{~T}^{-3}\right]\) (d) \(\left[\mathrm{MLT}^{-2}\right]\)

The dimensions of magnetic field in \(M, \mathrm{~L}, \mathrm{~T}\) and \(\mathrm{C}\) (Coulomb) are given as [AIEEE 2008] (a) \(\left[\mathrm{MLT}^{-1} \mathrm{C}^{-1}\right]\) (b) \(\left[\mathrm{MT}^{2} \mathrm{C}^{-2}\right]\) (c) \(\left[\mathrm{MT}^{-1} \mathrm{C}^{-1}\right]\) (d) \(\left[\mathrm{MT}^{-2} \mathrm{C}^{-1}\right]\)

Electric displacement is given by \(D=\varepsilon E\), Here, \(\varepsilon=\) electric permittivity \(E=\) electric field strength The dimensions of electric displacement are (a) \(\left[\mathrm{ML}^{-2} \mathrm{TA}\right]\) (b) \(\left[\mathrm{L}^{-2 \mathrm{~T}}{\underline{\phantom{xx}}}^{-1} \mathrm{~A}\right]\) (c) \(\left[\mathrm{L}^{-2} \mathrm{TA}\right]\) (d) None of these

The dimensional formula of magnetic flux is [BVP 2007] (a) \(\left[\mathrm{ML}^{0} \mathrm{~T}^{-2} \mathrm{~A}^{-1}\right]\) (b) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-1} \mathrm{~A}^{-1}\right]\) (c) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-1} \mathrm{~A}^{-2}\right]\) (d) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-2} \mathrm{~A}^{-1}\right]\)

The work done by a battery is \(W=\varepsilon \Delta q\), where \(\Delta q\) charge transferred by battery, \(\varepsilon=\) emf of the battery. What are dimensions of emf of battery? (a) \(\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{-2} \mathrm{~A}^{-2}\right]\) (b) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-3} \mathrm{~A}^{-2}\right]\) (c) \(\left[\mathrm{M}^{2} \mathrm{~L}^{0} \mathrm{~T}^{-3} \mathrm{~A}^{0}\right]\) (d) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-3} \mathrm{~A}^{-1}\right]\)

Dimensions of resistance in an electrical circuit, in terms of dimension of mass \(M\), of length \(L\), of time \(T\) and of current \(I\), would be \(\quad\) [UP SEE 2007] (a) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-3} \mathrm{I}^{-1}\right]\) (b) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-2}\right]\) [c) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-1} \mathrm{l}^{-1}\right]\) (d) \(\left[\mathrm{ML}^{2} \mathrm{~T}-3 \mathrm{I}^{-2}\right]\)

In the formula, \(a=3 b c^{2}, a\) and \(c\) have dimensions of electric capacitance and magnetic induction respectively. What are dimensions of \(b\) in MKS system? (a) \(\left[\mathrm{M}^{-3} \mathrm{~L}^{-2} \mathrm{~T}^{4} \mathrm{Q}^{4}\right]\) (b) \(\left[\mathrm{M}^{-3} \mathrm{~T}^{4} \mathrm{Q}^{4}\right]\) (c) \(\left[\mathrm{M}^{-3} \mathrm{~T}^{3} \mathrm{Q}\right]\) (d) \(\left[\mathrm{M}^{-3} \mathrm{~L}^{2} \mathrm{~T}^{4} \mathrm{Q}^{-4}\right]\)

Which of the following units denotes the dimensions \(\left[\mathrm{ML}^{2} / Q^{2}\right]\), where \(Q\) denotes the electric charge? [AIEEE 2006] (a) Henry (b) \(\mathrm{Hm}^{-2}\) (c) Weber (Wb) (d) \(\mathrm{Wbm}^{-2}\)

The dimensions of the power of lens are (a) \(\left[\mathrm{LT}^{-2}\right]\) (b) \(\left[\mathrm{M}^{0} \mathrm{~L}^{-1} \mathrm{~T}^{0}\right]\) (c) \(\left[\mathrm{M}^{0} \mathrm{~L}^{0} \mathrm{~T}^{0}\right]\) (d) None of these

The dimensions of permittivity \(\varepsilon_{0}\) are \(\quad[B V P 2006]\) (a) \(\left[\mathrm{M}^{-1} \mathrm{~L}^{-3} \mathrm{~A}^{2} \mathrm{~T}^{4}\right]\) (b) \(\left[\mathrm{M}^{-1} \mathrm{~L}^{3} \mathrm{~A}^{-2} \mathrm{~T}^{-4}\right]\) (c) \(\left[\mathrm{M}^{-1} \mathrm{~L}^{-1} \mathrm{~A}^{2} \mathrm{~T}^{2}\right]\) (d) \(\left[\mathrm{M}^{-1} \mathrm{~L}^{-3} \mathrm{~A}^{2} \mathrm{~T}^{-4}\right]\)

\(\left[\mathrm{ML}^{3} \mathrm{~T}^{-3} \mathrm{~A}^{-2}\right]\) is the dimensional formula of (a) Electric resistance (b) Capacity (c) Electric potential (d) Specific resistance

What is dimensional formula of thermal conductivity? [UP SEE 2006] (a) \(\left[\mathrm{MLT}^{-1} \theta^{-1}\right]\) (b) [MLT \(\left.^{-3} \theta^{-1}\right]\) (c) \(\left[\mathrm{M}^{2} \mathrm{LT}^{-3} \theta^{-2}\right]\) (d) \(\left[\mathrm{ML}^{2} \mathrm{~T}^{-2} \theta\right]\)

The concorde is the fastest airlines used for commercial service. It can cruise at 1450 mile per hour (about two times the speed of sound or in other words mach 2). What is it in \(\mathrm{m} / \mathrm{s}\) ? (a) \(644.4 \mathrm{~m} / \mathrm{s}\) (b) \(80 \mathrm{~m} / \mathrm{s}\) (c) \(40 \mathrm{~m} / \mathrm{s}\) (d) None of these

Which of the following is the most precise device for measuring length? (a) A vernier callipers with 20 divisions on the sliding scale (b) A screw gauge of pitch \(1 \mathrm{~mm}\) and 100 divisions on the circular scale (c) An optical instrument that can measure length to within a wavelength of light? (d) All are equally precise device for measuring length

Dimensions of potential energy are (a) \(\left[\mathrm{MLT}^{-1}\right]\) (b) \(\left[\mathrm{ML}^{2} \mathrm{~T}-2\right]\) (c) \(\left[\mathrm{ML}^{-1} \mathrm{~T}^{-2}\right]\) (d) \(\left[\mathrm{ML}^{-1} \mathrm{~T}^{-1}\right]\)

A student measures the thickness of a human hair by looking at it through a microscope of magnification 100\. He makes 20 observations and finds that the average width of the hair in the field of view of the microscope is \(3.5 \mathrm{~mm}\). The thickness of hair isfNCERT] (a) \(0.035 \mathrm{~mm}\) (b) \(0.04 \mathrm{~mm}\) (c) \(0.35 \mathrm{~mm}\) (d) \(0.40 \mathrm{~mm}\)

\(1 \mathrm{Wbm}^{-2}\) is equal to [UP SEE 2005] (a) \(10^{4}\) Gauss (b) \(4 \times 10^{-3}\) Gauss (c) \(10^{2}\) Gauss (d) \(10^{-4}\) Gauss