Units and Measurements Test

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Units and Measurements Test - 42

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Question 1
1 / -0

The foundations of dimensional analysis were laid down by

A
Gallileo

B
Newton

C
Fourier

D
Joule

Solution

The foundation of the dimensional analysis were laid down by FOURIER.
Question 2
1 / -0

The dimensional formula of wave number is

A
$$M^0L^0T^{-1}$$

B
$$M^0L^{-1}T^{0}$$

C
$$M^{-1}L^{-1}T^{0}$$

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

Solution

Wave number is given as =$$\dfrac{1}{\lambda} $$

$$\therefore$$ dimension is $$[M^0L^{-1}T^0]$$
Question 3
1 / -0

The dimensions of stress are equal to

A
Force

B
Pressure

C
Work

D
$$\dfrac{1}{Pressure}$$

Solution

We know that the stress is defined as :
$$Stress=\dfrac{Force}{Area}$$

It is same as that of pressure.

So, [Pressure] =[stress] = $$[ML^{-1}T^{-2}]$$

Question 4

1 / -0

Match the physical quantities given in Column I with suitable dimensions expressed in Column II.

Column I	Column II
a) Angular momentum	1) $${ M }^{ -1 }{ L }^{ 3 }{ T }^{ -2 }$$
b) Torque	2) $${ MLT }^{ -2 }$$
c) Gravitational constant	3) $${ ML }^{ 2 }{ T }^{ -2 }$$
d) Tension	4) $${ ML }^{ 2 }{ T }^{ -1 }$$

a-1, b-3, c-4, d-2

a-1, b-4, c-3 d-2

a-4, b-3, c-1, d-2

a-1, b-2, c-4, d-3

Solution

Angular momentum is given by $$\vec L =\vec r \times \vec p$$.

Thus, it's dimensions are $$[L]=L.MLT^{-1}=ML^{2}T^{-1}$$
Torque is given as $$\vec r \times \vec F$$.

Thus, dimensions are $$L.MLT^{-2}=ML^2T^{-2}$$.
Tension is force and hence its dimensions are $$MLT^{-2}$$.
Gravitational constant relationship is $$F=G\dfrac{Mm}{r^2}$$
Thus, $$[G]=\dfrac{[F][r^2]}{[Mm]}=\dfrac{MLT^{-2}.L^2}{M^2}=M^{-1}L^3T^{-2}$$
Thus, option C is correct.

Question 5
1 / -0

Which is not a dimensionless quantity is :

A
Moment of Momentum

B
Moment of force

C
Moment of inertia

D
All of the above

Solution

Hint: a dimensionless quantity is a quantity to which no physical dimension is assigned.
$${\textbf{Explanation:}}$$
A. Moment of Momentum $$= m v r$$
Dimension: $$ = M^{1} L^{2} T^{-1}$$

B. Moment of force $$= r \times F $$
Dimension: $$= M^{1} L^{2} T^{-2}$$

C. Moment of Inertia $$= Mr^{2} $$
Dimension: $$= M^{1} L^{2}$$

Hence, none of the quantity is dimensionless.

$${\textbf{Correct option: D}}$$
Question 6
1 / -0

A physicist performs an experiment and takes $$200$$ readings. He repeats the same experiment and now takes $$800$$ readings. By doing so :

A
the probable error remains same.

B
the probable error is four times.

C
the probable error is halved.

D
the probable error is reduced by a factor $$\dfrac{1}{4}$$.

Solution

More the number of experiments we do, improvement in our output will be achieved. And on an average our expected value of output will become more accurate. Therefore, probable error which is determined on average basis will get reduced by $$\dfrac{1}{4}$$ in this case.
Question 7
1 / -0

Consider the following two statements A and B. Identify the correct answer.
A) The quantity $$\dfrac { { e }^{ 2 } }{ { \epsilon }_{ 0 }ch } $$ is dimensionless.
B) $$\dfrac { 1 }{ \sqrt { { \mu }_{ 0 }{ \epsilon }_{ 0 } } } $$ has the dimensions of velocity and is numerically equal of velocity of light.

A
A is true but B is false

B
B is true but A is false

C
Both A and B are true

D
Both A and B are false

Solution

Dimensions of permittivity $${ \epsilon }_{ 0 }$$: $${ A }^{ 2 }{ T }^{ 4 }{ M }^{ -1 }{ L }^{ 3 }$$
Dimensions of $${ e }^{ 2 }$$ : $${ A }^{ 2 }{ T }^{ 2 }$$
Dimensions of $$c$$ : $$L{ T }^{ -1 }$$
Dimensions of $$h$$ : $$M{ L }^{ 2 }{ T }^{ -1 }$$
Therefore, dimensions of the given expression:
$$\dfrac { { A }^{ 2 }{ T }^{ 2 } }{ ({ A }^{ 2 }{ T }^{ 4 }{ M }^{ -1 }{ L }^{ 3 })\times (L{ T }^{ -1 })\times (M{ L }^{ 2 }{ T }^{ -1 }) } =1$$ which is dimensionless.

Hence, Statement A is true.
In statement B, the expression given is for the speed of light.Hence, it has dimensions of velocity.
Question 8
1 / -0

Choose the false statement from given statements.
I. Relative permittivity is dimensionless variable.
II. Angular displacement has neither units nor dimensions.
III. Refractive index is dimensionless variable.
IV. Permeability of vacuum is dimensional constant.

A
Only I and II

B
Only II

C
Only III

D
Only IV

Solution

Statements 1 and 3 are true because both these are ratios of two quantities with same dimensions, thus they are dimensionless.
Statement 4 is also correct because permeability of vacuum is a dimensionless constant with value equal to $$8.85\times 10^{-12}$$.
But angular displacements has unit (although no dimensions). It has the unit 'radian'.
Question 9
1 / -0

The dimensional formula for strain energy density is:

A
$$M^{1}L^{2}T^{-3}$$

B
$$M^{1}L^{2}T^{3}$$

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

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

Solution

Strain energy density $$ = \dfrac{Energy}{Volume} \\= \dfrac{M^{1}L^{2}T^{-2}}{L^{3}} \\= M^{1}L^{-1}T^{-2}$$

Question 10

1 / -0

Some physical constants are given in List - I and their dimensional formulae are given in List - II. Match the following :

List - I	List - II
A) Planck's constant	e) $$\left[ { ML }^{ -1 }{ T }^{ -2 } \right] $$
B) Gravitational constant	f) $$\left[ { ML }^{ 1 }{ T }^{ -1 } \right] $$
C) Bulk modulus	g) $$\left[ { ML }^{ 2 }{ T }^{ -1 } \right] $$
D) Linear momentum	h) $$\left[ { M}^{ -1 }{ L }^{ 3 }{ T }^{ -2 } \right] $$

A-h, B-g, C-e ,D-f

A-f, B-e, C-g, D-h

A-g, B-f, C-e, D-h

A-g, B-h, C-e, D-f

Solution

Planck's constant $$\displaystyle =\frac{\left ( energy \right )}{\left ( c/\lambda \right )}=\frac{ML^{2}T^{-2}}{\dfrac{LT^{-1}}{L}}=ML^{2}T^{-1}$$
Gravitational constant $$\displaystyle =\frac{force}{\left ( M_{1}M_{2} \right )}\times r^{2}=\frac{ML^{1}T^{-2}\times L^{2}}{M^{2}}$$$$=M^{-1}L^{3}T^{-2}$$

Bulk modulus $$\displaystyle =\frac{dp}{dv/v}=\frac{M^{1}L^{-1}T^{-2}}{L^{3}/L^{3}}=M^{1}L^{-1}T^{-2}$$

Linear momentum $$={ M }{ L }{ T }^{ -1 }$$

Hence, option D is correct.

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Units and Measurements Test - 42

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Units and Measurements Test - 42

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SHARING IS CARING

Question 1

Gallileo

Newton

Fourier

Joule

Solution

Question 2

$$M^0L^0T^{-1}$$

$$M^0L^{-1}T^{0}$$

$$M^{-1}L^{-1}T^{0}$$

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

Solution

Question 3

Force

Pressure

Work

$$\dfrac{1}{Pressure}$$

Solution

Question 4

a-1, b-3, c-4, d-2

a-1, b-4, c-3 d-2

a-4, b-3, c-1, d-2

a-1, b-2, c-4, d-3

Solution

Question 5

Moment of Momentum

Moment of force

Moment of inertia

All of the above

Solution

Question 6

the probable error remains same.

the probable error is four times.

the probable error is halved.

the probable error is reduced by a factor $$\dfrac{1}{4}$$.

Solution

Question 7

A is true but B is false

B is true but A is false

Both A and B are true

Both A and B are false

Solution

Question 8

Only I and II

Only II

Only III

Only IV

Solution

Question 9

$$M^{1}L^{2}T^{-3}$$

$$M^{1}L^{2}T^{3}$$

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

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

Solution

Question 10

A-h, B-g, C-e ,D-f

A-f, B-e, C-g, D-h

A-g, B-f, C-e, D-h

A-g, B-h, C-e, D-f

Solution

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