Units and Measurements Test

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Units and Measurements Test - 45

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Weekly Quiz Competition

Question 1
1 / -0

SI unit of a physical quantity whose dimensional formula is $${ M }^{ -1 }{ L }^{ -2 }{ T }^{ 4 }{ A }^{ 2 }$$ is :

A
$$ohm$$

B
$$volt$$

C
$$siemen$$

D
$$farad$$

Solution

Capacitance $$C=\dfrac{Q^2}{2E}=\dfrac{A^2T^2}{ML^2T^{-2}}=M^{-1}L^{-2}T^4A^2$$

Option D: farad is unit for capacitance.
Question 2
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A is a tank filled to its 75% with water, B is a weighing balance and C is a stone hung from a stand. If fig. 1 is correct, what do you expect to be the position of needle in fig. 2?

A

B

C

D
Question 3
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The dimensions of
$$\left ( \mu _{0}\varepsilon _{0} \right )^{-1/2}$$ are:

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

B
$$\left [ LT ^{-1}\right ]$$

C
$$\left [ L^{1/2} T^{1/2}\right ]$$

D
$$\left [ L^{1/2} T^{-1/2}\right ]$$

Solution

We know that the speed of an electromagnetic wave is given by :
$$c=\dfrac{1}{\sqrt{\mu _{0}\varepsilon _{0}}} $$

Therefore, $$(\mu_0\varepsilon_0)^{-1/2}$$ has the dimension the same as that of the speed.

$$\therefore dimension \,\,LT^{-1}$$
Question 4
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The velocity $$v$$ of a particle at time $$t$$ is given by $$v = at + \frac{b} {t + c}$$, where $$a, b\ and\ c$$ are constants.The dimensions of $$a, b\ and\ c$$ are respectively:

A
LT$$^{-2}$$, L and T

B
L$$^{2}$$, T and LT$$^{2}$$

C
LT$$^{2}$$, LT and L

D
L, LT and T$$^{2}$$

Solution

Two terms can be added or subtracted if their dimensions are same.
Thus $$c$$ has the dimension of time i.e. $$[T]$$
Dimension of $$v$$ and $$at$$ must be same.
Thus dimensions of $$[a] = \dfrac{[v]}{[t]} = \dfrac{[LT^{-1}]}{[T]} = [LT^{-2}]$$
Dimensions $$[b] = [v][t] = [LT^{-1}][T] = L$$
Question 5
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If the units of ML are doubled, then the unit of kinetic energy will become :

A
8 times

B
16 times

C
4 times

D
2 times

Solution

Kinetic energy is defined as $$E = \dfrac{1}{2}mv^2$$
Dimensions of kinetic energy $$E = [ML^2T^{-2}]$$
$$\therefore$$ $$\dfrac{E'}{E} = \dfrac{[(M')(L')^2(T)^{-2}]}{[ML^2T^{-2}]} = \dfrac{{[(2M) (2L)^2(T)^{-2}}]}{[ML^2T^{-2}]} = 8$$
Thus if the units of M and L are doubled, then the unit of kinetic energy will become $$8$$ times.
Question 6
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The dimension of the ratio of angular momentum to linear momentum is

A
$$ L^0 $$

B
$$ L^1 $$

C
$$ L^2 $$

D
$$MLT$$

Solution

Angular momentum is given by: $$L = r \times mv$$
Linear momentum is given by: $$p = mv$$
Ratio of angular momentum to the linear momentum is: $$\dfrac{L}{p} = \dfrac{r\times mv}{mv} = r$$
Unit of $$r$$ is meter. Hence, dimension of $$r$$ is $$[L^1]$$
Therefore, the dimension of the ratio of angular momentum and linear momentum is: $$\dfrac{L}{p} = [L^1]$$
Question 7
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$$ M^{-1}L^{-2}T^3\theta^1 $$ are the dimensions of :

A
coefficient of thermal conductivity

B
coefficient of viscosity

C
modulus of rigidity

D
thermal resistance

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).
Hence, the thermal resistance is ratio of temperature difference by power. Dimension of power is $$ML^{2}T^{-3}$$.
Therefore, unit of thermal resistance is $$kg^{-1} m^{-2}s^{3}K$$
Hence, the dimensional formula is $$[M^{-1} L^{-2} T^{3} \theta^{1}]$$
Question 8
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The number of significant figures in 3400 is :

A
4

B
1

C
3

D
2

Solution

Trailing zeros in a whole number are not significant.Hence $$3400$$ has 2 significant digits only.
Therefore, option D is correct.
Question 9
1 / -0

The string of a kite is 100 m long and it makes

an angle of $$60^{0}$$ with the horizontal. If there is no slack in

the string, find the height of the kite from the ground.

A
$$ 50 \sqrt{3} m$$

B
$$ 100 \sqrt{3} m$$

C
$$ 50 \sqrt{2} m$$

D
100 m
Question 10
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The dimensions of intensity of energy are :

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

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

C
$$ M{T}^{-3} $$

D
$$ ML^{-2}T^3 $$

Solution

The energy flowing per unit area per unit time is called the intensity of energy.
The intensity of energy is (by formula):
$$I = \dfrac{E}{At}$$
$$I = \dfrac{[M^1L^2T^{-2}]}{[L^2][T^1]}$$
$$I = [M^1L^0T^{-3}]$$
Hence, the dimensional formula of intensity of energy is $$[MT^{-3}]$$.