Units and Measurements Test

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Units and Measurements Test - 62

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Weekly Quiz Competition

Question 1
1 / -0

The dimensional formula of electric potential is

A
$$[ML^2T^{-3}A^{-1}]$$

B
$$[M^{-1}L^2T^{-2}A]$$

C
$$[M^{-1}L^2T^{-2}A^{-1}]$$

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

Solution

Electric potential,
$$\displaystyle V = \frac{W}{q} = \frac{[ML^2T^{-2}]}{[AT]} = [ML^2T^{-3}A^{-1}]$$
Question 2
1 / -0

Which of the following relation is dimensionally incorrect?

A
1 u = 931.5 MeV

B
1 u = 931.5 MeV/c$$^2$$

C
1 u = 1.67 x 10$$^{-27}$$kg

D
None of these

Solution

As, we know, $$E = mc^2$$
and energy released can be given as:
$$E=931MeV$$
for, $$m =1u$$
$$1u = 931.5 MeV/c^{2}$$ - Dimensionally correct [Dimension of mass]
$$1u=1.67\times 10^{-27}\ kg$$ - Dimensionally correct [Dimension of mass]
but, $$931.5 \ MeV$$ is having the dimension of energy so it can't be equated to dimension of mass or dimension of $$1u$$.
Hence, dimensionally correct data is given in $$option(A)$$
Hence, the correct option will be $$(A)$$
Question 3
1 / -0

The velocity of a particle (v) at an instant t is given by v =at + bt$$^{2}$$. The dimension of b is the

A
$$[L]$$

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

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

D
$$[LT^{-3}]$$

Solution

$$v = at + bt^2$$

$$[v] = [bt^2]$$ or $$[LT^{-1}] = [bT^2]$$ or $$[b] = [LT^{-3}]$$
Question 4
1 / -0

Dimensional formula of $$\Delta Q$$, heat supplied to the system is

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

B
$$[MLT^{-2}]$$

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

D
$$[MLT^1]$$

Solution

Heat supplied to a system is in the form of energy.
$$\therefore$$ Its dimensional formula is $$\displaystyle [ML^2T^{-2}]$$.
Question 5
1 / -0

The dimensional formula of physical quantity is $$[M^aL^bT^c]$$. Then that physical quantity is

A
surface tension if a = 1, b = 1, c = -2

B
force if a = 1, b = 1, c = 2

C
angular frequency if a = 0, b = 0, c = -1

D
spring constant if a = 1, b = -1, c = -2

Solution

1) Surface Tension $$[S] = \dfrac{Force [F]}{Length [L]}$$

So, $$S = \dfrac{[M^1 L^1 T^{-2}]}{[L^1]}$$

$$\therefore$$ Surface Tension $$= S = [M^1 L^0 T^{-2}]$$

2) Force $$[F] = $$ mass (m) $$\times $$ acceleration (a)

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

$$F = [M^1 L^1 T^{-2}]$$

3) Angular frequency $$ = w = \dfrac{red [d \phi ]}{\sec [dt]}$$

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

4) Spring constant $$[K] = \dfrac{Force \ applied [F]}{Extension \ of \ spring [x]}$$

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

$$K = [M^1 T^{-2}]$$

$$K = [M^1 L^0 T^{-2}]$$

$$\therefore$$ Option C is correct.
Question 6
1 / -0

The displacement of a progressive wave is represented by y = A sin($$\omega$$t - kx) where x is distance and t is time. The dimensions of $$\displaystyle \frac{\omega}{k}$$ are same as those of the

A
velocity

B
wave number

C
wavelength

D
frequency

Solution

$$y = A sin (\omega t - kx)$$
As $$(\omega t - kx)$$ represents an angle which is dimensionless, therefore
$$\displaystyle [\omega] = \frac{1}{[t]} = [T^{-1}]$$ and $$\displaystyle [k] = \frac{1}{[x]} = [L^{-1}]$$
$$\displaystyle \left[ \frac{\omega}{k} \right] = \frac{[T^{-1}]}{[L^{-1}]} = [LT^{-1}]$$ = velocity
Question 7
1 / -0

The mean length of an object is 5 cm. Which of the following measurements is most accurate. ?

A
4.9 cm

B
4.805 cm

C
5.25 cm

D
5.4 cm

Solution

To signify the mean length of an object comprising $$5$$ cm in dimension, it is better to present it with $$4.9$$ cm can be easily rounded up to $$5$$ cm.
Whereas, other options are either less than or greater than the given option.
Question 8
1 / -0

The physical quantity to be measured more accurately in the experimental determination of Young's modulus of a wire is

A
Radius

B
Length

C
Force

D
Extension

Solution

In the experimental determination of Young's modulus, the extension $$\delta L$$ in the wire is measured and using this Young's modulus is calculated by the relation :
$$Y = \dfrac{F \ L}{A \ \delta L}$$
Question 9
1 / -0

The dimensional formula for electric intensity is

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

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

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

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

Solution

Since $$E=\cfrac { F }{ q } ,\left[ E \right] =\cfrac { \left[ { M }^{ 1 }{ L }^{ 1 }{ T }^{ -2 } \right] }{ \left[ AT \right] } =\left[ { M }^{ 1 }{ L }^{ 1 }{ T }^{ 3 }{ A }^{ -1 } \right] $$
Question 10
1 / -0

Consider two cylindrical rods of identical dimensions, one of rubber and the other of steel. Both the rods are fixed rigidly at one end to the roof. A mass $$M$$ is attached to each of the free ends at the centre of the rods.

A
Both the rods will elongate but there shall be no perceptible change in shape

B
The steel rod will elongate and change shape but the rubber rod will only elongate

C
The steel rod will elongate without any perceptible change in shape, but the rubber rod will elongate and the shape of the bottom edge will change to an ellipse

D
The steel rod will elongate, without any perceptible change in shape, but the rubber rod will elongate with the shape of the bottom edge tapered to a tip at the centre

Solution

According to the diagram shown below in which a man $$AT$$ is attached at the center of each rod, then both rods will be elongated. But due to different elastic properties of the material the steel rod will elongate without making any perceptible change in shape, but the rubber rod will elongate with the shape of the bottom edge tapered to a tip at the centre.

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Units and Measurements Test - 62

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Units and Measurements Test - 62

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SHARING IS CARING

Question 1

$$[ML^2T^{-3}A^{-1}]$$

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

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

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

Solution

Question 2

1 u = 931.5 MeV

1 u = 931.5 MeV/c$$^2$$

1 u = 1.67 x 10$$^{-27}$$kg

None of these

Solution

Question 3

$$[L]$$

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

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

$$[LT^{-3}]$$

Solution

Question 4

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

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

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

$$[MLT^1]$$

Solution

Question 5

surface tension if a = 1, b = 1, c = -2

force if a = 1, b = 1, c = 2

angular frequency if a = 0, b = 0, c = -1

spring constant if a = 1, b = -1, c = -2

Solution

Question 6

velocity

wave number

wavelength

frequency

Solution

Question 7

4.9 cm

4.805 cm

5.25 cm

5.4 cm

Solution

Question 8

Radius

Length

Force

Extension

Solution

Question 9

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

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

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

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

Solution

Question 10

Both the rods will elongate but there shall be no perceptible change in shape

The steel rod will elongate and change shape but the rubber rod will only elongate

The steel rod will elongate without any perceptible change in shape, but the rubber rod will elongate and the shape of the bottom edge will change to an ellipse

The steel rod will elongate, without any perceptible change in shape, but the rubber rod will elongate with the shape of the bottom edge tapered to a tip at the centre

Solution

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