Units & Measurements — Page 12 | NEET Physics

Q111

A force defined by

F=\alpha t^2+\beta t

acts on a particle at a given time

t

. The factor which is dimensionless, if

\alpha

and

\beta

are constants, is:

A

\beta t / \alpha

B

\alpha t / \beta

C

\alpha \beta t

D

\alpha \beta / t

Correct Answer

Option B

Solution

The force acting on a particle at a given time $t$ is defined as $F = \alpha t^2 + \beta t$ , where $\alpha$ and $\beta$ are constants.

We need to determine which factor is dimensionless if $\alpha$ and $\beta$ are constants.

Using the principle of homogeneity of dimensions: $\begin{aligned} & [F] = [\alpha t^2] = [\beta t] \\ & [\alpha] = \dfrac{[F]}{[t^2]} \quad \text{and} \quad [\beta] = \dfrac{[F]}{[t]} \\ & \therefore \quad [\alpha][t] = [\beta] \\ & \therefore \quad \dfrac{\alpha t}{\beta} = \text{dimensionless} \end{aligned}$

Q112

The diameter of a spherical bob, when measured with vernier callipers yielded the following values :

3.33 \mathrm{~cm}, 3.32 \mathrm{~cm}, 3.34 \mathrm{~cm}, 3.33 \mathrm{~cm}

and

3.32 \mathrm{~cm}

. The mean diameter to appropriate significant figures is :

A

3.328 \mathrm{~cm}

B

3.3 \mathrm{~cm}

C

3.33 \mathrm{~cm}

D

3.32 \mathrm{~cm}

Correct Answer

Option C

Solution

\text{ } \begin{aligned} \text{ Mean diameter } & =\frac{\mathrm{d}_1+\mathrm{d}_2+\mathrm{d}_3+\mathrm{d}_4+\mathrm{d}_5}{5} \\ & =\frac{3.33+3.32+3.34+3.33+3.32}{5} \\ & =3.328 \approx 3.33 \end{aligned}

Q113

The mechanical quantity, which has dimensions of reciprocal of mass

(\mathrm{M}^{-1})

is :

A angular momentum

B coefficient of thermal conductivity

C torque

D gravitational constant

Correct Answer

Option D

Solution

Angular momentum

=\left[\mathrm{ML}^2 \mathrm{~T}^{-1}\right]

Coeff of thermal conductivity

=\left[\mathrm{MLT}^{-3} \mathrm{~K}^{-1}\right]

Torque

=\left[\mathrm{ML}^2 \mathrm{~T}^{-2}\right]

Gravitational constant

=\left[\mathrm{M}^{-1} \mathrm{~L}^3 \mathrm{~T}^{-2}\right]

So, gravitational constant has power of

(-1)

of

\mathrm{M}

.

Q114

The errors in the measurement which arise due to unpredictable fluctuations in temperature and voltage supply are :

A Personal errors

B Least count errors

C Random errors

D Instrumental errors

Correct Answer

Option C

Solution

Error arise due to unpredictable fluctuation in temperature and voltage supply are $\rightarrow$ random errors.

Q115

A metal wire has mass

(0.4 \pm 0.002) ~\mathrm{g}

, radius

(0.3 \pm 0.001) ~\mathrm{mm}

and length

(5 \pm 0.02) ~\mathrm{cm}

. The maximum possible percentage error in the measurement of density will nearly be :

A 1.3%

B 1.6%

C 1.4%

D 1.2%

Correct Answer

Option B

Solution

\rho=\frac{M}{V}

\rho=\frac{M}{\pi r^{2} \ell}

\frac{\Delta \rho}{\rho}=\frac{\Delta \mathrm{M}}{\mathrm{M}}+\frac{2 \Delta \mathrm{r}}{\mathrm{r}}+\frac{\Delta \ell}{\ell}

\frac{\Delta \rho}{\rho} \%=\left[\frac{0.002}{0.4}+\frac{2(0.001)}{(0.3)}+\frac{0.02}{5}\right] \times 100 \%

=\frac{1}{2} \%+\frac{2}{3} \%+\frac{2}{5} \%

=1.6 \%

Q116

The physical quantity that has the same dimensional formula as pressure is

A Coefficient of viscosity

B Force

C Momentum

D Young's modulus of elasticity

Correct Answer

Option D

Solution

Pressure

[P] = \left[ {{F \over A}} \right] = \left[ {{{ML{T^{ - 2}}} \over {{L^2}}}} \right] = [M{L^{ - 1}}{T^{ - 2}}]

And, dimensions of Young's modulus

[Y] = \left[ {{{Stress} \over {Strain}}} \right] = [M{L^{ - 1}}{T^{ - 2}}]

[Force]

= [ML{T^{ - 2}}]

[Momentum]

= [ML{T^{ - 1}}]

[Coefficient of viscosity]

= [M{L^{ - 1}}{T^{ - 1}}]

Q117

The percentage error in the measurement of g is : (Given that

g = {{4{\pi ^2}L} \over {{T^2}}}

,

L = (10\, \pm \,0.1)

cm,

T = (100\, \pm \,1)

s)

A 7%

B 2%

C 5%

D 3%

Correct Answer

Option D

Solution

Given

g = {{4{p^2}L} \over {{T^2}}}

Fractional error in value of g

{{\Delta g} \over g} = {{\Delta L} \over L} + 2{{\Delta T} \over T}

\Rightarrow {{\Delta g} \over g} \times 100 = {{\Delta L} \over L} \times 100 + 2{{\Delta T} \over T} \times 100

\Rightarrow {{\Delta g} \over g} \times 100 = {{0.1} \over {10}} \times 100 + 2 \times {1 \over {100}} \times 100 = 3\%

Hence, percentage error in measurement of g is 3%

Q118

The dimensions [MLT

-

2 A

-

2 ] belong to the

A Magnetic flux

B Self inductance

C Magnetic permeability

D Electric permittivity

Correct Answer

Option C

Solution

Dimensional formula of magnetic permeability is [MLT $-$ 2 A $-$ 2 ]

Q119

Plane angle and solid angle have

A Units but no dimensions

B Dimensions but no units

C No units and no dimensions

D Both units and dimensions

Correct Answer

Option A

Solution

Plane angle =

{{Arc} \over {Radius}} = {{[L]} \over {[L]}}\overset{{}}\longrightarrow

Unit = Radian

= [{M^0}{L^0}{T^0}]

Solid angle =

{{Area} \over {{{(Radius)}^2}}}\overset{{}}\longrightarrow

Unit = Steradian

= {{{L^2}} \over {{L^2}}} = [{M^0}{L^0}{T^0}]

$\therefore$ Both have units but no dimensions.

Q120

The area of a rectangular field (in m 2 ) of length 55.3 m and breadth 25 m after rounding off the value for correct significant digits is

A 138

\times

10 1

B 1382

C 1382.5

D 14

\times

10 2

Correct Answer

Option D

Solution

Area = Length $\times$ Breadth = 55.3 $\times$ 25 m 2 = 1382.5 m 2 = 14 $\times$ 10 2 m 2 (Rounding off of two significant figures)