Units and Measurements Test

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Units and Measurements Test - 49

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Weekly Quiz Competition

Question 1
1 / -0

If $C$ and $R$ denote capacitance and resistance, then dimensions of $CR$ will be

A
$\displaystyle \left [ M^{0}L^{0}TA^{0} \right ]$

B
$\displaystyle \left [ ML^{0}TA^{-2} \right ]$

C
$\displaystyle \left [ ML^{0}TA^{2} \right ]$

D
$\displaystyle \left [ ML^{0}T^{2}A^{-2} \right ]$

Solution

We know that value $RC$ gives time constant in an $R-C$ circuit and that $V(t)={V}_{0}{e}^{\frac{-t}{RC}}$
Exponents are always dimensionless, thus $[\dfrac{-t}{RC}]={M}^{0}{L}^{0}{T}^{0}{A}^{0}$ or $[RC]=[t]=[{M}^{0}{L}^{0}{T}^{1}{A}^{0}]$
Question 2
1 / -0

It takes time $8$ min for light to reach from sun to the earth surface. If speed of light is taken to be $3 \times 10^8$ m $s^{-1}$ , find the order of magnitude of distance from the sun to the earth in km.

A
$10^7 km$

B
$15\times10^8 km$

C
$10^8 km$

D
$10^9 km$

Solution

Time $t=8\ min=480\ sec$
Speed of light $c=3\times10^8\ ms^{-1}$
$Distance=t\times c=1.44\times10^{11}m=1.44\times10^8km$
So order of magnitude os distance is $10^8km$
Question 3
1 / -0

Write the dimensions of $\cfrac{d{\phi}_{B}}{dt}$ where $\phi_B$ is magnetic flux through a coil.

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

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

C
$[ M{ L }^{ 2 }{ A }^{ -2 }{ T }^{ -3 }]$

D
None of these

Solution

$\cfrac{d{\phi}_{B}}{dt}=$ [potential or EMF] $=\left[ M{ L }^{ 2 }{ A }^{ -1 }{ T }^{ -3 } \right]$
Question 4
1 / -0

The length, breadth and height of a glass slab are respectively $50 cm$ , $20 cm$ and $25 cm$ . Find the order of magnitude of volume of slab in Sl unit.

A
$10^{-2} m^3$

B
$10^{-4} m^3$

C
$10^{-3} m^3$

D
$10^{-3} cm^3$

Solution

$L=0.5 m , b=0.2 m ,h=0.25 m$
Volume $=$ length $\times$ breadth $\times$ height
Volume $=l\times b\times h=0.025 m^{3 }\ = \ 2.5\times 10^{-2}\ m^3$
So, the order of magnitude is $10^{-2} m^3$ .
Question 5
1 / -0

The radius of a hydrogen atom is $0.5\mathring{A}$ . Find the order of magnitude of volume of $1$ mole hydrogen in $m^3$ . Given that 1 mole of hydrogen has $6.02 \times 10^{23}$ hydrogen atoms.

A
$2\times10^{-4} m^3$

B
$10^{-3} m^3$

C
$10^{-7} m^3$

D
$10^{-2} m^3$

Solution

Volume of 1 hydrogen atom, $V=4\pi r^3/3; r=0.5\times10^{-10}m$
So, $V=0.5245\times10^{-30}m^3$
So, volume of 1 mole ( $6.02\times10^{23}$ ) is $V\times6.02\times10^{23}=3.157\times10^{-7}m^3$
So, order of magnitude is $10^{-7}m^3$ .
Question 6
1 / -0

Find the order of magnitude of the mass of the star, whose radius is $384 \times 10^6$ m and average density is $4 \times 10^3$ kg $m^{-3}$ :

A
30

B
29

C
24

D
21

Solution

Radius of star $r=384\times10^6m$
So, Volume of star is $V=4\pi r^3/3=2.371\times10^{26}m^3$
Density of star is $\rho=4\times10^3kgm^{-3}$
Mass of star is $M=V\times\rho\approx9.5\times10^{29}kg=0.95\times10^{30}kg$
So the mass is of order of $10^{30}kg$
Question 7
1 / -0

The electric potential existing in space is $V(x, y, z) = A (xy+ yz + zx)$ . Write the dimensional formula of $A$ :

A
$MT^{-2} I^{-1}$

B
$MT^{-3} I^{-1}$

C
$MT^{-2} I^{-2}$

D
$MT^{-3} I^{-2}$

Solution

Since $x, y$ and $z$ have units of length i.e. $m$ .
Thus Unit of A =unit of potential $\times m^{-2}$ = joules per coulomb $\times m^{-2}$
Dimension of joule $= ML^2 T^{-2}$
Dimension of coulomb $= I T$ where $I$ is the dimension of current.
Dimension of A = dimension of joule per coulomb $\times L^{-2}$
= $M L^2 T^{-2} (I T)^{-1} \times L^{-2} = M T^{-3} I^{-1}$
Question 8
1 / -0

The value due regard to significant figure $4.0\times 10^{-4}-2.5\times 10^{-6}$

A
$\displaystyle 4.0\times 10^{-4}$

B
$\displaystyle 3.975\times 10^{-4}$

C
$\displaystyle 3.9\times 10^{-4}$

D
none of the above

Solution

The value of this case is,
$4.0\times 10^{-4}-2.5\times 10^{-6}$
$=3.975\times 10^{-4}$ .
Question 9
1 / -0

Round off 2.0082 to four significant figures.

A
2.008

B
2.0082

C
2.009

D
2

Solution

$2.0082=2.008$ ,
Here the digit $2$ is less than $5$ so $2.008$ is the closest value in four significant figures.
Question 10
1 / -0

Dimension of relative density is -

A
$kg m^{-3}$

B
$ML^{-3}$

C
Dimensionless

D
$M^2L^{-6}$

Solution

Relative density is equal to the ratio of density of substance to density of water. Since both have the same dimensions, thus their ratio is dimensionless.

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Units and Measurements Test - 49

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Units and Measurements Test - 49

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Question 1

[M0L0TA0]\displaystyle \left [ M^{0}L^{0}TA^{0} \right ][M0L0TA0]

[ML0TA−2]\displaystyle \left [ ML^{0}TA^{-2} \right ][ML0TA−2]

[ML0TA2]\displaystyle \left [ ML^{0}TA^{2} \right ][ML0TA2]

[ML0T2A−2]\displaystyle \left [ ML^{0}T^{2}A^{-2} \right ][ML0T2A−2]

Solution

Question 2

107km10^7 km107km

15×108km15\times10^8 km15×108km

108km10^8 km108km

109km10^9 km109km

Solution

Question 3

[ML2A−1T−3][ M{ L }^{ 2 }{ A }^{ -1 }{ T }^{ -3 }] [ML2A−1T−3]

[ML3A−1T−2][ M{ L }^{ 3 }{ A }^{ -1 }{ T }^{ -2 }] [ML3A−1T−2]

[ML2A−2T−3][ M{ L }^{ 2 }{ A }^{ -2 }{ T }^{ -3 }] [ML2A−2T−3]

None of these

Solution

Question 4

10−2m310^{-2} m^310−2m3

10−4m310^{-4} m^310−4m3

10−3m310^{-3} m^310−3m3

10−3cm310^{-3} cm^310−3cm3

Solution

Question 5

2×10−4m32\times10^{-4} m^32×10−4m3

10−3m310^{-3} m^310−3m3

10−7m310^{-7} m^310−7m3

10−2m310^{-2} m^310−2m3

Solution

Question 6

30

29

24

21

Solution

Question 7

MT−2I−1MT^{-2} I^{-1}MT−2I−1

MT−3I−1MT^{-3} I^{-1}MT−3I−1

MT−2I−2MT^{-2} I^{-2}MT−2I−2

MT−3I−2MT^{-3} I^{-2}MT−3I−2

Solution

Question 8

4.0×10−4\displaystyle 4.0\times 10^{-4}4.0×10−4

3.975×10−4\displaystyle 3.975\times 10^{-4}3.975×10−4

3.9×10−4\displaystyle 3.9\times 10^{-4}3.9×10−4

none of the above

Solution

Question 9

2.008

2.0082

2.009

2

Solution

Question 10

kgm−3 kg m^{-3} kgm−3

ML−3 ML^{-3} ML−3

Dimensionless

M2L−6 M^2L^{-6} M2L−6

Solution

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$\displaystyle \left [ M^{0}L^{0}TA^{0} \right ]$

$\displaystyle \left [ ML^{0}TA^{-2} \right ]$

$\displaystyle \left [ ML^{0}TA^{2} \right ]$

$\displaystyle \left [ ML^{0}T^{2}A^{-2} \right ]$

$10^7 km$

$15\times10^8 km$

$10^8 km$

$10^9 km$

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

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

$[ M{ L }^{ 2 }{ A }^{ -2 }{ T }^{ -3 }]$

$10^{-2} m^3$

$10^{-4} m^3$

$10^{-3} m^3$

$10^{-3} cm^3$

$2\times10^{-4} m^3$

$10^{-3} m^3$

$10^{-7} m^3$

$10^{-2} m^3$

$MT^{-2} I^{-1}$

$MT^{-3} I^{-1}$

$MT^{-2} I^{-2}$

$MT^{-3} I^{-2}$

$\displaystyle 4.0\times 10^{-4}$

$\displaystyle 3.975\times 10^{-4}$

$\displaystyle 3.9\times 10^{-4}$

$kg m^{-3}$

$ML^{-3}$

$M^2L^{-6}$