Units and Measurements Test

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Units and Measurements Test - 73

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Weekly Quiz Competition

Question 1
1 / -0

The dimensions of $$\frac{1}{{{\mu _0}{ \in _0}}}$$ is (where symbols have their usual menaing)

A
$$\left[ {{L^2}{T^{ - 2}}} \right]$$

B
$$[A]$$

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

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

Solution

Velocity of light in vacuum is $${\frac{1}{{\sqrt {\left( {{\mu _0}{ \in _0}} \right)} }}}$$
$$\left[ {L{T^{ - 1}}} \right] = \left[ {\frac{1}{{\sqrt {\left( {{\mu _0}{ \in _0}} \right)} }}} \right]$$
$$\left[ {{L^2}{T^{ - 2}}} \right] = \left[ {\frac{1}{{\left( {{\mu _0}{ \in _0}} \right)}}} \right]$$
Hence,
option $$(A)$$ is correct answer.
Question 2
1 / -0

The dimensional formula of $$ \frac{GM_{e}}{gRe^{2}}$$ is ---------

A
$$M^{1}L^{1}T^{2} $$

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

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

D
$$M^{0}L^{1}T^{0} $$

Solution

Dimensional formula $$\dfrac{Gm_2}{gR^2_e}$$

for $$G=M^{-1}L^3T^{-2}$$
$$M_e=M$$
$$g=LT^{-2}$$
$$R_e=L$$

$$\therefore \dfrac{m^{-1}L^3T^{-2}}{LT^{-2}\cdot L}M$$
$$\Rightarrow M^0LT^0$$
$$\Rightarrow L$$
$$\Rightarrow M^0L^1T^0$$.
Question 3
1 / -0

The resultant of two equal forces acting at right angles to each other is $$1414$$ dyne. Find the magnitude of either force.

A
$$960$$

B
$$1000$$

C
$$1200$$

D
$$1414$$

Solution

Given,
$$F_1=F_2=F(say)$$
$$\theta=90^0$$
$$R=1414dyne$$
Resultant force,
$$R=\sqrt{F_1^2+F_2^2+2F_1F_2cos\theta}$$
$$1414dyne=\sqrt{F^2+F^2+2F^2cos90^0}=\sqrt{2F^2}$$
$$1414dyne=\sqrt{2}F$$
$$F=\dfrac{1414}{\sqrt{2}}=999.84 \approx 1000dyne$$ (by rounding off)
The correct option is B.
Question 4
1 / -0

If $$m,e,{\varepsilon _0},$$ and $$c$$ denote mass electron, charge of electron Plank's constant and speed of light, respectively. The dimensions of $$\frac{{m{e^4}}}{{\varepsilon _0^2{h^3}c}}$$ are

A
$$\left[ {{M^0}{L^0}{T^{ - 1}}} \right]$$

B
$$\left[ {{M^0}{L^{ - 1}}{T^{ - 1}}} \right]$$

C
$$\left[ {{M^0}L{T^{ - 3}}} \right]$$

D
$$\left[ {{M^2}{L^{ - 1}}{T^0}} \right]$$

Solution
Question 5
1 / -0

Which of the following pair does not have identical dimension ?

A
Moment of inertia and moment of force

B
Angular momentum and plank's constant

C
Impulse and momentum

D
Work and torque.

Solution

Planck's constant, symbolized h, relates the energy in one quantum (photon) of electromagnetic radiation to the frequency of that radiation.

$$h=\dfrac Ev=\dfrac{[ML^2T^{-2}]}{[T^{-1}]}=[ML^2T^{-1}]$$

Angular momentum $$=$$ Moment of inertia $$\times$$ Angular velocity (Angular momentum)

$$=[ML^2][T^{-1}]=[ML^2T^{-1}]$$

So, here we can see that both the Planck's constant and Angular momentum have the same dimensions.
Question 6
1 / -0

The dimensions of $$\cfrac{1}{2}{\in }_{0}{E}^{2}$$, where $${\in }_{0}$$ is permittivity of free space and $$E$$ is electric field, is:

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

B
$$\left[ M{ L }^{ 2 }{ T }^{ -2 } \right] $$

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

D
$$\left[ M{ L }^{ 2 }{ T }^{ -1 } \right] $$
Question 7
1 / -0

Out of the following the more precise value is .....

A
$$1.200 \times 10 ^ { 3 } k m$$

B
$$1.20 \times 10 ^ { 3 } k m$$

C
$$1.2 \times 10 ^ { 3 } k m$$

D
$$1200 k m$$

Solution
Question 8
1 / -0

The sum of the numbers $$436.32$$, $$227.244$$ and $$0.301$$ in appropriate significant figures is:

A
$$663.821$$

B
$$664$$

C
$$663.8$$

D
$$663.82$$

Solution

$$\begin{array}{l}\text{ According to the rule of addition more decimal places }\\\text { will be preferred in case of addition. }\end{array}$$

$$\begin{array}{l}=436.32+227.244+0.301\\=663.865\mathrm{}\end{array}$$

$$\therefore \text{the most appropriate number out of all the options is option C i.e. 663.8}$$
Question 9
1 / -0

Which of the following pairs of physical quantities have same dimension$$?$$

A
force and work

B
torque and energy

C
torque and power

D
force and torque

Solution

we know$$,$$ force $$=$$ mass $$×$$ acceleration
energy $$=$$ Workdone $$=$$ force $$×$$ displacement
power $$=$$ energy/time
torque $$=$$ force $$×$$ perpendicular distance between force and axis of rotation$$.$$
now$$,$$ by these formula$$,$$ find dimension of all terms$$.$$
e.g., dimension of force $$= [MLT⁻²]$$
dimension of energy $$= [ML²T⁻²]$$
dimension of power $$= [ML²T⁻³]$$
dimension of torque $$= [ML²T⁻²]$$
Here it is clear that $$,$$ dimension of energy and dimension of torque are equal$$.$$
Hence,
option $$(b)$$ is correct$$.$$
Question 10
1 / -0

An equation is given by $$\dfrac{d}{dt}[\int\vec{v}.\vec{dS}]$$-$$k[\vec{v}.\vec{\theta}]$$. If $$\vec{v}$$ represents velocity, $$\vec{ds}$$ is small dosplacement, $$t$$ is time and a is acceleratation. The dimension of $$k$$ is

A
$$[T]$$

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

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

D
$$[L]$$

Solution

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Units and Measurements Test - 73

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Units and Measurements Test - 73

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Question 1

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

$$[A]$$

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

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

Solution

Question 2

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

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

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

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

Solution

Question 3

$$960$$

$$1000$$

$$1200$$

$$1414$$

Solution

Question 4

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

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

$$\left[ {{M^0}L{T^{ - 3}}} \right]$$

$$\left[ {{M^2}{L^{ - 1}}{T^0}} \right]$$

Solution

Question 5

Moment of inertia and moment of force

Angular momentum and plank's constant

Impulse and momentum

Work and torque.

Solution

Question 6

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

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

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

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

Question 7

$$1.200 \times 10 ^ { 3 } k m$$

$$1.20 \times 10 ^ { 3 } k m$$

$$1.2 \times 10 ^ { 3 } k m$$

$$1200 k m$$

Solution

Question 8

$$663.821$$

$$664$$

$$663.8$$

$$663.82$$

Solution

Question 9

force and work

torque and energy

torque and power

force and torque

Solution

Question 10

$$[T]$$

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

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

$$[L]$$

Solution

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