Units and Measurements Test

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Units and Measurements Test - 69

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Question 1
1 / -0

If the dimension of a physical quantity is given by $$[M^aL^bT^c]$$ then the physical quantity will be

A
acceleration, if a = 1, b =-1, c =-2

B
velocity, if a = 1, b =0, c =-1

C
pressure, if a = 1, b =-1, c =-2

D
force, if a = 0, b =-1, c =-2

Solution

Pressure, $$P=\dfrac{Force}{Area}=\dfrac{Mass\times Acceleration}{Area}$$

$$[P]=\dfrac{ML^1T^{-2}}{L^2}=[M^1L^{-1}T^{-2}]=[M^aL^bT^c]$$

$$a=1\,,b=-1\,,c=-2$$
Question 2
1 / -0

The equation of state of some gases can be expressed as $$\left(P+\dfrac{a}{V^{2}}\right)\left(V-b\right)=RT$$. Here P is the pressure, $$V$$ is the volume, $$T$$ is the absolute temperature and $$a,b,R$$ are constant. The dimensions of $$a$$ are:

A
$$ML^{5}T^{-2}$$

B
$$ML^{-1}T^{-2}$$

C
$$M^{0}L^{3}T^{0}$$

D
$$M^{0}L^{6}T^{0}$$

Solution

Dimension, $$\dfrac{a}{{{V}^{2}}}$$ is the same as pressure dimension is $$N{{m}^{-2}}$$
$$ \dfrac{a}{{{V}^{2}}}\,=\dfrac{a}{{{\left( {{m}^{3}} \right)}^{2}}}=N{{m}^{-2}} $$
$$ a=N{{m}^{4}}=\,\left[ M{{L}^{1}}{{T}^{-2}} \right]\left[ {{M}^{o}}{{L}^{4}}{{T}^{o}} \right]=\left[ M{{L}^{5}}{{T}^{-2}} \right] $$
Hence, unit of $$a$$ is $$\left[ M{{L}^{5}}{{T}^{-2}} \right]$$
Question 3
1 / -0

Find the maximum possible percentage error in the measurement of force on an object(on mass m) travelling at velocity v in a circle of radius r, if m=(4.0 plus minus 0.1)kg, v=(10 plus minus 0.1)m/s and r=(8.0 plus minus 0.2)m

A
( 50 ± 1) N

B
( 50 ± 2.5) N

C
( 50 ± 1.5) N

D
( 50 ± 3.5) N

Solution

Force$$(f)=\cfrac{mv^2}{r}=\cfrac{4\times(10)^2}{8}=50\\ \therefore\cfrac{\Delta F}{F}=\cfrac{\Delta m}{m}+2(\cfrac{\Delta V}{V})+(\cfrac{\Delta r}{r})\\ \quad=\cfrac{0.1}{4}+2\cfrac{(0.1)}{10}+\cfrac{0.8}{8}\\ \therefore \cfrac{\Delta F}{F}=0.07$$
Percentage error in $$F=25\%+2+2.5\%$$
$$\therefore$$ Percentage error in $$F=7\%$$
Eror in $$\Delta F$$ in $$F=F\times0.07\\ \quad=0\times0.07$$
$$\therefore$$ Error in $$\Delta F$$ in $$F=3.5N$$
$$\therefore F=(50\pm3.5)N$$
Question 4
1 / -0

Choose the incorrect statements out of the following:-

A
Every measurement by any measuring instrument has some errors

B
Every calculated physical quantity that is based on measured values has some error

C
A measurement can have more accuracy but less precision and vice-versa

D
The percentage error is different from relative error.

Solution

$$ \text { We have relative error as } $$
$$ \text { Relative error = } \frac{\text { measured value - real value }}{\text { real value }} $$ $$ \text { We have persentage error as } $$
$$ \text { persentage error = } \frac{\text { measured value - real value }}{\text { real value }} $$
$$ \begin{array}{l} \text { So we get a Relation } \\ \qquad\text { Relative error }=\frac{\text { Percent error }}{100} \end{array} $$
Question 5
1 / -0

The dimensional formula $$[ML^1T^{-3}]$$ is more closely associated with

A
power

B
energy

C
intensity

D
velocity gradient

Solution
Question 6
1 / -0

The dimensional formula for thermal resistance is

A
$$\left [ ML^{2}T^{-3}K^{-1} \right ]$$

B
$$\left [ ML^{2}T^{-2}A^{-1} \right ]$$

C
$$\left [ ML^{2}T^{-3}K^{-2} \right ]$$

D
$$\left [ M^{-1}L^{-2}T^{3}K \right ]$$

Solution

Thermal resistance is a heat property and a measurement of a temperature difference by which an object or material resists a heat flow (heat per unit time or thermal resistance).

$$Thermal\, Resisatnce$$ $$=\dfrac{Temperature\,difference}{Thermal\,current}=\dfrac{\Delta T}{Rate\,of\,flow\,of\,heat}$$

$$\implies \dfrac{\Delta T}{\dfrac{\Delta Q}{\Delta t}}=\dfrac{[K]}{[ML^2T^{-3}]}=[M^{-1}L^{-2}T^3K]$$

The dimensional formula for thermal resistance is $$[M^{-1}L^{-2}T^3K]$$
Question 7
1 / -0

Dimensional formula of self inductance is:

A
$$M L T^{-2} A^{-2}$$

B
$$M L^2 T^{-1} A^{-2}$$

C
$$M L^2 T^{-2} A^{-2}$$

D
$$M L^2 T^{-2} A^{-1}$$

Solution

The magnetic flux is given by
$$\phi=LI$$
The dimensional formula of self inductance, $$[L]=\dfrac{[BA]}{[I]}$$. . . . . .(1)
Force, $$f=qvB$$
$$[M^1L^1T^{-2}]=[TA].[L^1T^{-1}].[B]$$
$$[B]=[M^1T^{-2}A^{-1}]$$
$$[BA]=[M^1L^2T^{-2}A^{-1}]$$
From equation (1), we get
$$[\phi]=[M^1L^2T^{-2}A^{-2}]$$
The correct option is C.
Question 8
1 / -0

Suppose an attractive nuclear force acts between two protons, which may be written as $$F=\dfrac { { Ae }^{ -kr } }{ { r }^{ 2 } } $$. The dimensional formula for $$\dfrac {A}{K}$$ is

A
$${M}^{1}{L}^{2}{T}^{2}$$

B
$${M}^{1}{L}^{3}{T}^{-1}$$

C
$${M}^{1}{L}^{4}{T}^{1}$$

D
$${M}^{1}{L}^{4}{T}^{-2}$$

Solution

Given,

$$F=\dfrac{Ae^{-kr}}{r^2}$$

$$[F]=\dfrac{[A]}{[r^2]}$$

$$\implies {[A]}=[F][r^2]$$

$$\implies [A]=[MLT^{-2}]{[L^2]}$$

$$[A]=[ML^3T^{-2}]$$

We know that $$Kr$$ is unitless.

Therefore, S.I unit of $$K=\dfrac 1m$$

Then,

$$[K]=\dfrac{1}{[L]}=[L^{-1]}$$

So,

$$\dfrac {[A]}{[k]}=\dfrac{[ML^3T^{-2}]}{[L^{-1}]}=[M^1L^4T^{-2}]$$
Question 9
1 / -0

The dimension formula of torque is

A
$$M{L^2}$$

B
$$MLT$$

C
$$MLT^2$$

D
$$ML^2T^{-2}$$

Solution

Torque, $$\tau=r\times F=rma$$
The dimensional formula of torque,
$$[\tau]=[r][m][a]$$
$$[\tau]=[L][M][LT^{-2}]$$
$$[\tau]=[ML^2T^{-2}]$$
The correct option is D.
Question 10
1 / -0

Find the dimensions of inductance

A
$$ML^{2}T^{-2}A^{-2}$$

B
$$ML^{2}T^{-1}A^{-1}$$

C
$$MLT^{-1}A^{-1}$$

D
$$MLTA^{-2}$$

Solution

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Units and Measurements Test - 69

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Units and Measurements Test - 69

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Question 1

acceleration, if a = 1, b =-1, c =-2

velocity, if a = 1, b =0, c =-1

pressure, if a = 1, b =-1, c =-2

force, if a = 0, b =-1, c =-2

Solution

Question 2

$$ML^{5}T^{-2}$$

$$ML^{-1}T^{-2}$$

$$M^{0}L^{3}T^{0}$$

$$M^{0}L^{6}T^{0}$$

Solution

Question 3

( 50 ± 1) N

( 50 ± 2.5) N

( 50 ± 1.5) N

( 50 ± 3.5) N

Solution

Question 4

Every measurement by any measuring instrument has some errors

Every calculated physical quantity that is based on measured values has some error

A measurement can have more accuracy but less precision and vice-versa

The percentage error is different from relative error.

Solution

Question 5

power

energy

intensity

velocity gradient

Solution

Question 6

$$\left [ ML^{2}T^{-3}K^{-1} \right ]$$

$$\left [ ML^{2}T^{-2}A^{-1} \right ]$$

$$\left [ ML^{2}T^{-3}K^{-2} \right ]$$

$$\left [ M^{-1}L^{-2}T^{3}K \right ]$$

Solution

Question 7

$$M L T^{-2} A^{-2}$$

$$M L^2 T^{-1} A^{-2}$$

$$M L^2 T^{-2} A^{-2}$$

$$M L^2 T^{-2} A^{-1}$$

Solution

Question 8

$${M}^{1}{L}^{2}{T}^{2}$$

$${M}^{1}{L}^{3}{T}^{-1}$$

$${M}^{1}{L}^{4}{T}^{1}$$

$${M}^{1}{L}^{4}{T}^{-2}$$

Solution

Question 9

$$M{L^2}$$

$$MLT$$

$$MLT^2$$

$$ML^2T^{-2}$$

Solution

Question 10

$$ML^{2}T^{-2}A^{-2}$$

$$ML^{2}T^{-1}A^{-1}$$

$$MLT^{-1}A^{-1}$$

$$MLTA^{-2}$$

Solution

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