Units and Measurements Test

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Units and Measurements Test - 50

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Weekly Quiz Competition

Question 1
1 / -0

Write the dimensional formula of charge.

A
$$\left [ AT \right ]$$

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

C
$$\left [ AT^{-2} \right ]$$

D
$$\left [ AT^{2} \right ]$$

Solution

Since current is rate of flow of charge, i.e. $$I=\dfrac { q }{ t } $$. Therefore $$q=It$$. Hence, $$[q]=[AT]$$.
Question 2
1 / -0

Calorie is a unit of heat or energy whose value is $$4.2J$$, where $$J=1\;kgm^2s^{-2}$$. If one uses a unit system in which units of mass, length and time are taken as $$\alpha\;kg,\;\beta\;metre$$ and $$\gamma\;second$$ respectively, then the value of calorie in this system will be

A
$$\;4.2\alpha^{-1}\beta^{-2}\gamma^{2}$$

B
$$\;4.2\alpha^1\beta^2\gamma^{-2}$$

C
$$\;\alpha^{1}\beta^{-2}\gamma^{-2}$$

D
$$\;\alpha^{-1}\beta^{2}\gamma^{-2}$$

Solution

$$\because\,1\;cal=4.2\;J\;\;\;\therefore\,1\;cal=4.2\;kgm^2s^{-2}$$

$$\;\;\;\;\;\;\;\;\;\;\;\;=(4.2\,\alpha^{-1}\beta^{-2}\gamma^2)(\alpha kg)(\beta m)^2(\gamma s)^{-2}$$
Question 3
1 / -0

Round off following number upto four significant figures.
$$45.689$$

A
$$45.689$$

B
$$45.69$$

C
$$45.68$$

D
$$45.7$$

Solution

The given number has $$5$$ significant figures.
As $$9>5$$, thus the digit $$8$$ gets changed to $$9$$ after rounding off upto four significant figures.
$$\therefore$$ After rounding off, $$45.689$$ becomes $$45.69$$
Question 4
1 / -0

Write down the number of significant figures in the following value.
$$1200 N$$

A
$$4$$

B
$$3$$

C
$$2$$

D
$$1$$

Solution

Zeroes after non zero digit and before decimal are not counted as significant.
Therefore, (C) has two significant figures in $$1200 N$$.
Question 5
1 / -0

In the relation $$F=a\;sin\;k_1x+b\;sin\;k_2t$$, the units (SI) of $$\displaystyle\frac{k_1}{k_2}$$ is ......... .
($$F,x\,\&\,t$$ have usual meanings)

A
$$s/m$$

B
$$sm$$

C
$$st$$

D
$$s/t$$

Solution

The argument of sin must be dimensionless.
Hence, unit of $${ k }_{ 1 }$$ must be $${ m }^{ -1 }$$ and that of $${ k }_{ 2 }$$ must be $${ s }^{ -1 }$$.
Hence, unit of $$\dfrac { { k }_{ 1 } }{ { k }_{ 2 } } $$ will be $$s/m$$.
Question 6
1 / -0

In two different systems of units an acceleration is represented by the same number, whilest a velocity is represented by numbers in the ratio $$1\,\colon\,3$$. The ratios of unit of length and time are

A
$$\;\displaystyle\frac{1}{3},\displaystyle\frac{1}{9}$$

B
$$\;\displaystyle\frac{1}{9},\displaystyle\frac{1}{3}$$

C
$$\;1,1$$

D
$$\;\displaystyle\frac{1}{3},\displaystyle\frac{1}{3}$$

Solution

Given, $$\dfrac{[v_1]}{[v_2]}=\dfrac{1}{3}$$
Here, $$[a_1]=[a_2]$$ or $$[v_1T_1^{-1}]=[v_2T_2^{-1}]$$ ( as $$a=\dfrac{dv}{dt}$$)

$$\dfrac{T_1}{T_2}=\dfrac{v_1}{v_2}=\dfrac{1}{3}$$

$$\dfrac{[L_1T_1^{-1}]}{[L_2T_2^{-1}]}=\dfrac{1}{3}$$ as $$ [v]=[LT^{-1}]$$

$$\dfrac{L_1}{L_2}=\dfrac{1}{3}\times \dfrac{[T_1]}{[T_2]}=1/3 \times 1/3=1/9$$
Question 7
1 / -0

A force $$\displaystyle F $$ is given by $$ F = at + bt^2 $$, where $$ t $$ is time. The dimensions of $$a$$ and $$b$$ are :

A
$$ [M\,L\,T^{-3}] $$ and $$ [M\,L\,T^{-4}] $$

B
$$ [M\,L\,T^{-4}] $$ and $$ [M\,L\,T^{-3}] $$

C
$$ [M\,L\,T^{-1}] $$ and $$ [M\,L\,T^{-2}] $$

D
$$ [M\,L\,T^{-2}] $$ and $$ [M\,L\,T^{0}] $$

Solution

Dimensions of force are $$MLT^{-2}$$. Thus, both $$at$$ and $$bt^2$$ must have units of force.
Thus, $$|a|T=MLT^{-2}$$ or $$|a|=MLT^{-3}$$
Also, $$|b|T^2=MLT^{-2}$$ or $$|b|=MLT^{-4}$$
Question 8
1 / -0

From the following pairs, choose the pair that does not have identical dimensions ?

A
Impulse and momentum

B
Work and torque

C
Moment of inertia and moment of force

D
Angular moment and Planck's constant

Solution

Impulse is change in momentum ($$p_2-p_1$$).
Thus, they have same dimensions.
Work is dot product of Force and displacement while Torque is the cross product of Force and radius vector.
Thus they have same dimensions (of Fr = $$MLT^{-2}\times L = ML^2T^{-2}$$)
Angular moment and Planck's constant also have the same dimensions because Angular moment = Momentum $$\times $$ length. Planck's constant is Momentum $$\times$$ Wavelength (because $$p=\dfrac{h}{\lambda}$$)
Thus they have same dimensions.
Moment of Inertia has dimensions $$ML^2$$ and moment of force, i.e. Torque has dimensions $$ML^2T^{-2}$$, thus different.
Question 9
1 / -0

The ratio of the dimensions of Planck's constant and that of inertia has the dimension of ?

A
Frequency

B
Velocity

C
Angular momentum

D
Time

Solution

The plank's constant has the dimensions of
$$h=[M^1L^2T^{-1}]$$

The dimension of the inertia is
$$I=[M^1L^2]$$

The ratio of the dimension of the plank's constant and the inertia is
$$\dfrac hI=T^{-1}$$

It is the same as the dimension of frequency.
Question 10
1 / -0

The dimensions of universal gravitational constant $$G$$ are :

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

B
$$ [M\,L^3\,T^{-2}] $$

C
$$ [M^{-1}\,L^3\,T^{-2}] $$

D
$$ [M^{-1}\,L\,T^{-2}] $$

Solution

$$\textbf{Hint:}$$ Dimensional formula of force is $$ML{{T}^{-2}}$$ .
$$\textbf{Solution:}$$
$$\textbf{Step 1: Calculating the gravitational constant.}$$
We know that Gravitational force exerted on a body of mass m is given by,
$$F=\dfrac{GMm}{{{r}^{2}}}$$
$$\Rightarrow G=\dfrac{F{{r}^{2}}}{Mm}$$
$$\textbf{Step 2: Calculating dimensions of universal gravitational constant.}$$
$$G=\dfrac{[ML{{T}^{-2}}][{{L}^{2}}]}{[{{M}^{2}}]}$$
$$\Rightarrow G=[{{M}^{-1}}{{L}^{3}}{{T}^{-2}}]$$
$$\textbf{Thus, option (C) is correct.}$$

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Units and Measurements Test - 50

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Units and Measurements Test - 50

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SHARING IS CARING

Question 1

$$\left [ AT \right ]$$

$$\left [ AT^{-1} \right ]$$

$$\left [ AT^{-2} \right ]$$

$$\left [ AT^{2} \right ]$$

Solution

Question 2

$$\;4.2\alpha^{-1}\beta^{-2}\gamma^{2}$$

$$\;4.2\alpha^1\beta^2\gamma^{-2}$$

$$\;\alpha^{1}\beta^{-2}\gamma^{-2}$$

$$\;\alpha^{-1}\beta^{2}\gamma^{-2}$$

Solution

Question 3

$$45.689$$

$$45.69$$

$$45.68$$

$$45.7$$

Solution

Question 4

$$4$$

$$3$$

$$2$$

$$1$$

Solution

Question 5

$$s/m$$

$$sm$$

$$st$$

$$s/t$$

Solution

Question 6

$$\;\displaystyle\frac{1}{3},\displaystyle\frac{1}{9}$$

$$\;\displaystyle\frac{1}{9},\displaystyle\frac{1}{3}$$

$$\;1,1$$

$$\;\displaystyle\frac{1}{3},\displaystyle\frac{1}{3}$$

Solution

Question 7

$$ [M\,L\,T^{-3}] $$ and $$ [M\,L\,T^{-4}] $$

$$ [M\,L\,T^{-4}] $$ and $$ [M\,L\,T^{-3}] $$

$$ [M\,L\,T^{-1}] $$ and $$ [M\,L\,T^{-2}] $$

$$ [M\,L\,T^{-2}] $$ and $$ [M\,L\,T^{0}] $$

Solution

Question 8

Impulse and momentum

Work and torque

Moment of inertia and moment of force

Angular moment and Planck's constant

Solution

Question 9

Frequency

Velocity

Angular momentum

Time

Solution

Question 10

$$ [M\,L\,T^{-2}] $$

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

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

$$ [M^{-1}\,L\,T^{-2}] $$

Solution

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