Units and Measurements Test

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Units and Measurements Test - 76

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Weekly Quiz Competition

Question 1
1 / -0

In the equation $${ S }_{ nth }=u+\dfrac { a }{ 2 } \left( 2n-1 \right) $$, the letters have their usual meanings. The dimensional formula of $$S_{nth}$$ is

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

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

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

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

Solution

The Dimensional Formula of $${S}_{nth}$$ is same as that of the Dimensional formula of $$u$$ (2 variables with same units can be added)

So Dimensional Formula of $${S}_{nth}$$ = DImensional Formula of $${u}$$ = $$[M^{0}{L}{T}^{-1}]$$
Question 2
1 / -0

A physical quantity x depends on quantities y and Z as follows : x= Ay+B tan (Cz), where A, B and C are constants . Which of the following do not have same dimensions ?

A
x and B

B
C and $$z^{ -1 }$$

C
y and B/A

D
x and A

Solution
Question 3
1 / -0

Dimensions of rate of change of flux are equivalent to those of

A
voltage

B
current

C
1/charge

D
charge
Question 4
1 / -0

The dimension of the ratio of magnetic flux and the resistance is equal to that of:

A
induced emf

B
charge

C
inductance

D
current

Solution

Dimension of magnetic flux $$=[ML^2T^{-2}A^{-1}]$$

Dimension of resistance $$=ML^2T^{-3}A^{-2}]$$

Therefore,

$$\dfrac{[Magnetic\,\,flux]}{[Resistance]}=\dfrac{[ML^2T^{-2}A^{-1}]}{[ML^2T^{-3}A^{-2}]}=[AT]=Charge$$
Question 5
1 / -0

If pressure is (P), velocity (V), and time (T) are taken as the fundamental quantities, then the dimensional formula of force is

A
$$\left[ P ^ { 1 } V ^ { 1 } T ^ { 1 } \right]$$

B
$$\left[ P ^ { 1 } V ^ { 2 } T ^ { 1 } \right]$$

C
$$\left\lceil P ^ { 1 } V ^ { 1 } T ^ { 2 } \right\rceil$$

D
$$\left[ P ^ { 1 } V ^ { 2 } T ^ { 2 } \right]$$

Solution

$$Force=P^xV^yT^z$$

$$[MLT^{-2}]=[MLT^{-2}L^{-2}]^x[LT^{-1}]^y[T]^z$$

Comparing the dimensions on both side, we get,

$$x=1$$, $$y=2$$, $$z=2$$

So, $$Force=[P^1V^2T^2]$$
Question 6
1 / -0

In a system of units if force (F), acceleration (A) and time, (T) are taken as fundamental units then the dimensional formula of energy is -

A
$$FA^{2}T$$

B
$$FAT^{2}$$

C
$$F^{2}AT$$

D
FAT

Solution

Consider that the energy is expressed as:
$$E=KF^aA^bT^c$$

Substituting the dimensional formulas:
$$[ML^2T^{−2}]=[MLT^{−2}]^a[LT^{−2}]^b[T]^c$$

$$[ML^2T^{−2}]=[M^aL^{a+b}T^{−2a−2b+c}]$$

$$\Rightarrow a=1,a+b=2\Rightarrow b=1$$

and $$−2a−2b+c=−2\Rightarrow c=2$$

$$\therefore E=KFAT^2$$
Question 7
1 / -0

The dimensions of the ratio of angular momentum to linear momentum:

A
$$\left[ { M }^{ 1 }{ L }^{ -3 }{ T }^{ 0 } \right] $$

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

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

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

Solution
Question 8
1 / -0

The term $$(1/2)pv^{ 2 }$$ occurs in Bernoulli's equation , with $$\beta $$ being the density of fluid and V its speed . The dimensions of this term are

A
$$\left[ { M }^{ -1 }{ L }^{ 5 }{ T }^{ 2 } \right] $$

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

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

D
$$\left[ { M }^{ -1 }{ L }^{ 9 }{ T }^{ -2 } \right] $$

Solution
Question 9
1 / -0

Two resistance are measured in ohm and is given as:-
$${ R }_{ 1 }=3\Omega \neq 1%$$
$${ R }_{ 2 }=6\Omega \neq 2%$$
When they are connectrd in parallel, the percentage error in equivalent resistance is

A
$$3.33\%$$

B
$$4.5\%$$

C
$$0.67\%$$

D
$$1.33\%$$

Solution

$$\text { Given: } R_{1}=3 \Omega \pm 0 \cdot 1 \quad R_{2}=6 \Omega \pm 0.2$$

$$\begin{array}{l}\text { For parallel } \\\frac{1}{R_{e q}}=\frac{1}{R_{1}}+\frac{1}{R_{2}}=\frac{1}{3}+\frac{1}{6}=\frac{2+1}{6}=\frac{3}{6}=\frac{1}{2}\end{array}$$

$$\begin{array}{l}R_{e q}=2 \Omega \\\text { Differentiating eq }^{n} \text { (i) both side } \\-\frac{d R_{e q}}{R_{e q}^{2}}=-\frac{d R_{1}}{R_{1}^{2}}-\frac{d R_{2}}{R_{2}^{2}}\end{array}$$

$$\begin{array}{l}\text { Error is always added }\\\qquad \begin{aligned}\frac{d R_{e q}}{R_{e q}^{2}} &=\frac{d R_{1}}{R_{1}^{2}}+\frac{d R_{2}}{R_{2}^{2}} \\\frac{d R_{e q}}{R_{e q}}&=\left(\frac{d R_{1}}{R_{1}^{2}}+\frac{d R_{2}}{R_{2}^{2}}\right)R_{0}\mathrm{~V}\\&=\left(\frac{0.1}{3^{2}}+\frac{0.2}{6^{2}}\right){2}\\&=\left(\frac{0.1}{9}+\frac{0.2}{36}\right){2}\end{aligned}\end{array}$$

$$\begin{aligned}\frac{d \operatorname{Req} x}{\operatorname{Reg}} \times100\% &=\left(\frac{4+0.2}{36}\right)^{2} \times 100 \% \\&=\frac{0.6}{36}\times 2 \times 100 \\&=\frac{100}{3}\\&=3.33\%\end{aligned}$$
Question 10
1 / -0

If the radius of the earth were to shrink by 1% its mass remaining the same, the acceleration due to gravity on the earth's surface would

A
Decreased by 2%

B
Increased by 2%

C
Remain unchanged

D
Increased by 1%

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Re-Attempt Weekly Quiz Competition

Units and Measurements Test - 76

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Units and Measurements Test - 76

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Question 1

$$\left[ { ML }^{ 0 }T \right] $$

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

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

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

Solution

Question 2

x and B

C and $$z^{ -1 }$$

y and B/A

x and A

Solution

Question 3

voltage

current

1/charge

charge

Question 4

induced emf

charge

inductance

current

Solution

Question 5

$$\left[ P ^ { 1 } V ^ { 1 } T ^ { 1 } \right]$$

$$\left[ P ^ { 1 } V ^ { 2 } T ^ { 1 } \right]$$

$$\left\lceil P ^ { 1 } V ^ { 1 } T ^ { 2 } \right\rceil$$

$$\left[ P ^ { 1 } V ^ { 2 } T ^ { 2 } \right]$$

Solution

Question 6

$$FA^{2}T$$

$$FAT^{2}$$

$$F^{2}AT$$

FAT

Solution

Question 7

$$\left[ { M }^{ 1 }{ L }^{ -3 }{ T }^{ 0 } \right] $$

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

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

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

Solution

Question 8

$$\left[ { M }^{ -1 }{ L }^{ 5 }{ T }^{ 2 } \right] $$

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

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

$$\left[ { M }^{ -1 }{ L }^{ 9 }{ T }^{ -2 } \right] $$

Solution

Question 9

$$3.33\%$$

$$4.5\%$$

$$0.67\%$$

$$1.33\%$$

Solution

Question 10

Decreased by 2%

Increased by 2%

Remain unchanged

Increased by 1%

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