Units and Measurements Test

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Units and Measurements Test - 27

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Weekly Quiz Competition

Question 1
1 / -0

An athlete's coach told him that muscle times speed equals power.
Then according to him the dimensions of muscle are

A
$[MLT^{-2}]$

B
$[ML^2T^{-2}]$

C
$[MLT^{+2}]$

D
$[MLT^{-3}]$

Solution

Let us represent the dimension of muscle as X and we are given $X \times speed=power$ .

$X \times speed = power = \dfrac{Work}{Time}$ $=\dfrac{\left[ML^{2}T^{-2}\right]}{\left[T\right]}$

Now, $X=\dfrac{\left[ML^{2}T^{-3}\right]}{\left[LT^{-1}\right]}$ .

The dimension of muscle is $\left[MLT^{-2}\right]$ .
Question 2
1 / -0

A quantity $X$ is given by $\epsilon_0 L\dfrac {\Delta V}{\Delta t}$ , where $\epsilon_0$ is the permittivity of fee space, $L$ is length, $\Delta V$ is a potential difference and $\Delta t$ is a time interval. The dimensional formula for $X$ is the same as that of

A
resistance

B
charge

C
voltage

D
current

Solution

Dimensional formula for $\epsilon_0 = [M^{-1} L^{-3} T^{4} I^{2}]$
Dimensional formula for $L = [L]$
Dimensional formula for $\Delta V = [M L^2 T^{-3} I^{-1}]$
Dimensional formula for $\Delta t = [T^{1}]$
Therefore, dimensional formula for $X$ is: $\dfrac{ [M^{-1} L^{-3} T^{4} I^{2}] [L] [M L^2 T^{-3} I^{-1}] }{ [T^{1}] }=[I]$
Therefore, its the same as that of current.
Question 3
1 / -0

The correct statement about Poisson's ratio is:

A
Its unit is $N/m^2$

B
It is dimensionless

C
Its unit is Newton

D
Its dimensions are $MLT^{-2}$

Solution

When we apply forces on a body along the length in the opposite directions which are equal in magnitude, it undergoes extension at the same time and contraction along the perpendicular direction.
Within the elastic limits, the ration of lateral strain to the longitudinal strain is called Poisson's ratio and being a ratio it is a dimensionless quantity.
Question 4
1 / -0

The dimensions of quantity $\dfrac{1}{\mu_0} (\vec{E} \times \vec{B})$ are:
$[\mu_0=$ permeability of free space, $\displaystyle E^1=$ electric field strength, $\displaystyle B^1=$ magnetic field induction $]$

A
$[MT^{-3}]$

B
$[ML^2T^{-3}]$

C
$[ML^{-1}T^{-1}]$

D
$[MT^{-3}A^2]$

Solution

$\dfrac { \left[ E \right] \left[ B \right] }{ { \left[ { \mu }_{ 0 } \right] } } =\dfrac { \left[ ML{ T }^{ -3 }{ I }^{ -1 } \right] \left[ M{ T }^{ -2 }{ I }^{ -1 } \right] }{ \left[ ML{ T }^{ -2 }{ I }^{ -2 } \right] }$ $={ M }^{ 1+1-1 }{ L }^{ 1-1 }{ T }^{ -3-2+2 }{ I }^{ -1-1+2 }$ $=M{ T }^{ -3 }$
Question 5
1 / -0

The dimensional formula of latent heat is identical to that of

A
internal energy

B
angular momentum

C
gravitational potential

D
electric potential

Solution

Latent heat is given by:
$Q=mL$
$\Rightarrow L=\displaystyle \frac {Q}{m}$ i.e Energy per unit mass.
a) internal energy has the units of energy
b) angular momentum has the units Joule-sec, i.e Energy.second
c) Gravitational potential is the potential energy per unit mass, i.e Energy per mass
d) Electric potential is the potential energy per unit charge.
Question 6
1 / -0

The dimensional formula of $\dfrac {L}{R}$ is same as that of (where $L$ is inductance and $R$ is resistance)

A
(frequency) $^2$

B
time period

C
frequency

D
(time period) $^2$

Solution

Self inductance is given by: $\displaystyle v(t)=L\dfrac {di}{dt}$
From this we have: $\displaystyle L=v\dfrac {dt}{di}$

$\displaystyle \dfrac {v}{di}$ has the dimensions of $R$

So, $L=Rdt$ hence $\displaystyle \dfrac {L}{R}=dt$ , same units as that of time.
Question 7
1 / -0

Which of the following combinations has dimensions?

A
$\dfrac{rpv}{\eta}$

B
$\mu_0\in_0c^2$

C
$\dfrac{t}{RC}$

D
$\dfrac{\sigma}{\lambda_mT}$

Solution

All the first three options are dimensionless quantity except the last one.
Since $\left[ \sigma \right] =\left[ { M }{ T }^{ -3 }{ K }^{ -4 } \right] ,\quad \left[ { \lambda }_{ m } \right] =\left[ L \right] \quad \& \quad \left[ T \right] =\left[ K \right]$
Question 8
1 / -0

How many significant figures are there in $0.030100\times 10^6$ ?

A
1

B
3

C
5

D
7

Solution

As per the following rules of significant numbers:
1) ALL non-zero numbers (1,2,3,4,5,6,7,8,9) are ALWAYS significant.
2) ALL zeroes between non-zero numbers are ALWAYS significant.
3) Zeroes placed after other digits but behind a decimal point are significant 4)Zeroes placed before other digits are not significant.
Hence in given number i.e. $0.030100 \times 10^6$ can be further simplified as $30100.00$
According to the rule 1, 3 and rule 4, the total significant numbers are 5.
Question 9
1 / -0

The dimensional formula for relative density is

A
$[ML^{-3}]$

B
$[M^0L^{-3}]$

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

D
$[M^0L^0T^0]$

Solution

Relative density is the ratio of the density of a substance to the density of a given reference material.
Dimension of Relative density $=\left[ { M }^{ 0 }{ L }^{ 0 }{ T }^{ 0 } \right]$
Question 10
1 / -0

The ratio of the dimensions of Planck's constant and that of the moment of intertia is the dimensions of

A
time

B
frequency

C
angular momentum

D
velocity

Solution

Planck constant, $h=[ML^2T^{-1}]$
Moment of inertia, $I=[ML^2T^{0}]$
So, $\dfrac { \left[ h \right] }{ \left[ I \right] } =\dfrac { \left[ { M }^{ 1 }{ L }^{ 2 }{ T }^{ -1 } \right] }{ \left[ { M }^{ 1 }{ L }^{ 2 }{ T }^{ 0 } \right] } =\left[ { M }^{ 0 }{ L }^{ 0 }{ T }^{ -1 } \right] \Rightarrow frequency$

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Units and Measurements Test - 27

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Units and Measurements Test - 27

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Question 1

[MLT−2] [MLT^{-2}] [MLT−2]

[ML2T−2] [ML^2T^{-2}] [ML2T−2]

[MLT+2] [MLT^{+2}] [MLT+2]

[MLT−3][MLT^{-3}][MLT−3]

Solution

Question 2

resistance

charge

voltage

current

Solution

Question 3

Its unit is N/m2 N/m^2 N/m2

It is dimensionless

Its unit is Newton

Its dimensions are MLT−2 MLT^{-2} MLT−2

Solution

Question 4

[MT−3] [MT^{-3}] [MT−3]

[ML2T−3] [ML^2T^{-3}] [ML2T−3]

[ML−1T−1] [ML^{-1}T^{-1}] [ML−1T−1]

[MT−3A2] [MT^{-3}A^2] [MT−3A2]

Solution

Question 5

internal energy

angular momentum

gravitational potential

electric potential

Solution

Question 6

(frequency)2^22

time period

frequency

(time period)2^22

Solution

Question 7

rpvη \dfrac{rpv}{\eta} ηrpv​

μ0∈0c2 \mu_0\in_0c^2 μ0​∈0​c2

tRC \dfrac{t}{RC} RCt​

σλmT \dfrac{\sigma}{\lambda_mT} λm​Tσ​

Solution

Question 8

1

3

5

7

Solution

Question 9

[ML−3][ML^{-3}][ML−3]

[M0L−3][M^0L^{-3}][M0L−3]

[M0L0T−1][M^0L^0T^{-1}][M0L0T−1]

[M0L0T0][M^0L^0T^0][M0L0T0]

Solution

Question 10

time

frequency

angular momentum

velocity

Solution

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$[MLT^{-2}]$

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

$[MLT^{+2}]$

$[MLT^{-3}]$

Its unit is $N/m^2$

Its dimensions are $MLT^{-2}$

$[MT^{-3}]$

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

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

$[MT^{-3}A^2]$

(frequency) $^2$

(time period) $^2$

$\dfrac{rpv}{\eta}$

$\mu_0\in_0c^2$

$\dfrac{t}{RC}$

$\dfrac{\sigma}{\lambda_mT}$

$[ML^{-3}]$

$[M^0L^{-3}]$

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

$[M^0L^0T^0]$