Units and Measurements Test

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Units and Measurements Test - 72

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Weekly Quiz Competition

Question 1
1 / -0

The accuracy in the measurement of the diametre of a hydrogen atom as $${ 1.06\times 10 }^{ -10 }$$ m is

A
$$0.01$$

B
$${ 106\times 10 }^{ -10 }$$

C
$$\dfrac { 1 }{ 106 } $$

D
$${ 0.01\times 10 }^{ -10 }$$

Solution
Question 2
1 / -0

The diameter of wire measured with a varnier calliper with keast count 0.1 mm is Scm, then the percentage error in measurement is

A
$$0.2\%$$

B
$$2\%$$

C
$$5\%$$

D
$$0.5\%$$
Question 3
1 / -0

Of the following quantities , which one has dimensions different from the remaining three?

A
Energy per unit volume

B
Force per unit area

C
Product of voltage and charge per unit volume

D
Angular momentum

Solution

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

Force per unit area $$=\dfrac{Force}{Area}=\dfrac{MLT^{-2}}{L^2}=ML^{-1}T^{-2}$$

Product of the charge per unit volume and voltage $$=\dfrac{Work}{Charge}\times \dfrac{Charge}{Volume}=\dfrac{Work}{Volume}$$$$=\dfrac{ML^2T^{-2}}{L^3}=ML^{-1}T^{-2}$$

Angular momentum per unit mass $$=\dfrac{M^1L^2T^{-1}}{M}=L^2T^{-1}$$

So correct answer is option (D).
Question 4
1 / -0

If error in measurement of radius of a sphere is $$1\%$$, what will be the error in measurement volume?

A
$$1\%$$

B
$$\dfrac{1}{3}\%$$

C
$$3\%$$

D
None of these

Solution

$$Volume=\dfrac { 4πR^3 }{ 3 } $$

$$\dfrac { ΔV }{ V } \times 100=3\dfrac { ∆R }{ R } \times 100$$

$$=3\times \dfrac { 1 }{ 100 } \times 100$$

$$=3$$

∴Percentage error in volume is $$3$$%
Question 5
1 / -0

Which of the following measurements is most accurate?

A
$$0.005 mm$$

B
$$5.00 mm$$

C
$$50.00 mm$$

D
$$5.0 mm$$

Solution
Question 6
1 / -0

The dimensions of latent heat are -

A
$$M ^ { 0 } L ^ { 2 } T ^ { - 2 }$$

B
$$M ^ { 0 } L ^ { 1 } T ^ { - 2 }$$

C
$$M ^ { 2 } L ^ { 0 } T ^ { - 2 }$$

D
$$M ^ { 0 } L ^ { - 2 } T ^ { - 2 }$$

Solution

$$Latent\, heat (L)=\dfrac{Heat}{Mass}$$ . . . . . . (1)
The dimensional formula of mass $$=[M^1L^0T^0]$$. . . . (2)
Also, the dimensions of heat $$=$$ dimensions of energy $$=$$ dimensions of work
Since, $$work = force\times displacement$$ . . . . (3)
And, the dimensional formula of,
Displacement $$= [M^0L^1T^0] $$. . . (4)
Force $$= m \times a = [M^1L^1T^{-2}]$$ . . . (5)
On substituting equation (4) and (5) in equation (3) we get,
Work $$=[M^1L^1T^{-2}]\times [L^{1}]$$
Therefore, the dimensions of work or heat $$=[M^1L^2T^{-2}]$$ . . . . (6)
On substituting equation (2) and (6) in equation (1) we get,
Or, $$L = \dfrac{[M^1L^2T^{-2}] }{ [M^1L^0T^0]}=[M^0L^2T^{-2}]$$

Therefore, latent heat is dimensionally represented as $$[M^0L^2T^{-2}]$$
Question 7
1 / -0

If pressure $$P$$, area $$A$$ and length $$L$$ are taken to be fundamental quantities in a system of units, then energy has dimensional formula.

A
$$P^{1}A^{1}L^{1}$$

B
$$P^{1}A^{3/2}L^{0}$$

C
$$P^{1}A^{0}L^{3}$$

D
Cannot take $$P, A$$ and $$L$$ as fundamental quantities

Solution

We know that
$$P = \dfrac {P}{A} = \dfrac {MLT^{-2}}{L^{2}}$$
$$= ML^{-1} T^{-2}$$
$$A = L^{2}$$
$$L = L$$
Let $$E = P^{x} A^{y} L^{z}$$
$$E = ML^{2}T^{-2}$$
$$= (ML^{-1}T^{-2})^{x} (L^{2})^{y} (L)^{2}$$
$$= M^{x} L^{-x + 2y + z} T^{-2x}$$
$$x = 1, -2x = -2$$
$$x = 1$$
$$-x + 2y + z = 2$$
$$z = 0, 2y = 2 + 1$$
$$y = \dfrac {3}{2}$$
$$\therefore E = P^{1} A^{3/2} L^{0}$$.
Question 8
1 / -0

A uniform rod undergoes a longitudinal strain of $$2$$. find percentage change in its volume if poisson's ratio of the material is $$0.50$$

A
0.4%

B
0.7%

C
0%

D
0.3%

Solution

We know that realtion between moduler of elasticity $$\left( \varepsilon \right),$$ possion's ratio $$\left( u\right)$$ and balls modules elasticity $$\left(h\right).$$
Given by,
$$\Rightarrow \varepsilon =3h\left( 1-2u\right)$$
here given $$u=0.3$$
$$h\rightarrow \alpha$$
and we also know $$\rightarrow h=\dfrac{\dfrac{-\triangle p}{\triangle v}}{v}$$
If $$h$$ tends to infinite then $$\triangle v$$ tends to zero.
$$\%$$ change in volume will be zero.
Hence, solve.
Question 9
1 / -0

The error in the measurement of length of a simple pendulum is $$0.1\%$$ and error in the time period is $$2 \%$$. The possible maximum error in the quantity having dimensional formula $$LT^{-2}$$ is

A
$$1.1 \%$$

B
$$2.2\%$$

C
$$4.1\%$$

D
$$6.1\%$$

Solution

Given,
Error $$\%$$ in time $$=2\%=\dfrac{2}{100}=\dfrac{1}{50}=2\%$$
Error $$\%$$ in length $$=0.1\%$$
Find $$\%$$ error in $$\dfrac{1}{12}$$
$$\therefore$$ as $$\dfrac{\triangle A}{A}=\left( \dfrac{\triangle 2}{2}\right) +\left( \dfrac{\triangle Q}{Q}\right)$$ if $$A=\dfrac{t}{Q}$$
Thus,
$$\Rightarrow \%$$ error $$=\left( \dfrac{\triangle L}{L}\right) +2\left( \dfrac{\triangle T}{T}\right)$$
$$=\left( 2\%\right) +2\left( .1\%\right)$$
$$=2\%+.2\%$$
$$=2.2\%$$
Hence, the answer is $$2.2\%.$$
Question 10
1 / -0

Which of the following is a dimensional constant?

A
Refractive index

B
Possions ratio

C
Stress

D
Gravitaional constant

Solution

Hint:- Check the dimensional formula of all given quantities

Explanation :-
A) Refractive index:- It is the ratio of velocity of light in a vacuum to its velocity in specified medium.
$$\mu=\dfrac{c}{v}$$
$$[\mu]=\dfrac{[c]}{[v]}=\dfrac{[M^0L^0T^{-1}]}{[M^0L^1T^{-1}]}$$
Hence, refractive index is dimensionless quantity.
B) Passions ratio:- The ratio of the proportional decreases in a lateral measurement to the proportional increases in length in a sample of the material that is elastically stretched.
Hence, it is dimensionless quantity.
C) Stress:- It is the ratio of force to the area.
$$s=\dfrac{F}{A}$$
$$[]=\dfrac{[MLT^{-2}]}{[L^2]}=[ML^{-1}T^{-2}]$$
It is not a dimensionless quantity. But it is not a constant value.
D) Gravitational constant:- It is a universal constant, denoted by $$G$$.
$$G=6.674\times 10^{-11}m^3/kgs^2$$
Dimensional value, $$[G]=[M^{-1}L^3T^{-2}]$$

The correct option is D.

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Units and Measurements Test - 72

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Units and Measurements Test - 72

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Question 1

$$0.01$$

$${ 106\times 10 }^{ -10 }$$

$$\dfrac { 1 }{ 106 } $$

$${ 0.01\times 10 }^{ -10 }$$

Solution

Question 2

$$0.2\%$$

$$2\%$$

$$5\%$$

$$0.5\%$$

Question 3

Energy per unit volume

Force per unit area

Product of voltage and charge per unit volume

Angular momentum

Solution

Question 4

$$1\%$$

$$\dfrac{1}{3}\%$$

$$3\%$$

None of these

Solution

Question 5

$$0.005 mm$$

$$5.00 mm$$

$$50.00 mm$$

$$5.0 mm$$

Solution

Question 6

$$M ^ { 0 } L ^ { 2 } T ^ { - 2 }$$

$$M ^ { 0 } L ^ { 1 } T ^ { - 2 }$$

$$M ^ { 2 } L ^ { 0 } T ^ { - 2 }$$

$$M ^ { 0 } L ^ { - 2 } T ^ { - 2 }$$

Solution

Question 7

$$P^{1}A^{1}L^{1}$$

$$P^{1}A^{3/2}L^{0}$$

$$P^{1}A^{0}L^{3}$$

Cannot take $$P, A$$ and $$L$$ as fundamental quantities

Solution

Question 8

0.4%

0.7%

0%

0.3%

Solution

Question 9

$$1.1 \%$$

$$2.2\%$$

$$4.1\%$$

$$6.1\%$$

Solution

Question 10

Refractive index

Possions ratio

Stress

Gravitaional constant

Solution

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