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If u1 and u2 are the units of a physical quantity and n1 and n2 are its n..

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Explanation

The questions relate to the concepts of measurement units and conversions between different units of length which are important in physics.

Question 1

Concepts

Units, numerical value, dimensions, relation between numerical value and unit

Explanation

If a physical quantity has a numerical value n1​ and unit u1​, and the same quantity is expressed as n2​ and u2​, then the product of numerical value and unit must be equal (because the physical quantity is same):

n1​×u1​=n2​×u2​

Rearranging:

n2​n1​​=u1​u2​​

(Note the units u1​ and u2​ are symbols for unit sizes, not numerical numbers, but this relation explains how numerical values change with changes in units.)

Step-By-Step Solution

Step 1

Start from defining the value of the physical quantity with two pairs (n1​,u1​) and (n2​,u2​): Q=n1​u1​=n2​u2​

Step 2

Divide both sides by n2​u1​: n2​n1​​=u1​u2​​

Step 3

Therefore option 4: n2​n1​​=u1​u2​​ is correct.

Final Answer

Option 4: n2​n1​​=u1​u2​​

Question 2

Concepts

Units of length: femto (fermi), micron, angstrom, nanometre, conversion of length units

Explanation

1 fermi is defined as a femtometer (fm), that is 10−15 meter.

Let’s convert other units to meters to compare:

  • 1 micron = 10−6 m
  • 1 angstrom = 10−10 m
  • 1 nanometre = 10−9 m

Now, check each option:

Option 1: 10−9 micron = 10−9×10−6 m = 10−15 m = 1 fermi

Option 2: 10−5 angstrom = 10−5×10−10 m = 10−15 m = 1 fermi

Option 3: 10−3 nanometre = 10−3×10−9 m = 10−12 m, which is not equal to fermi.

Step-By-Step Solution

Step 1

Convert 1 fermi to meters: 1 fermi=10−15 m

Step 2

Check option 1: 10−9 micron=10−9×10−6m=10−15m=1 fermi

Step 3

Check option 2: 10−5 angstrom=10−5×10−10m=10−15m=1 fermi

Step 4

Check option 3: 10−3 nanometre=10−3×10−9m=10−12m=1 fermi

Final Answer

Options 1 and 2 are correct:

  1. 10−9 micron
  2. 10−5 angstrom unit

Since the question expects only one answer, if choosing one, option 1 or 2 both are correct, but usually option 1 is preferred.

Question 3

Concepts

Length conversions, astronomical units: angstrom, fermi, parsec, light year

Explanation

Convert each length into meters and compare:

  1. 108 angstrom =108×10−10m=10−2m=0.01m

  2. 106 fermi =106×10−15m=10−9m

  3. 0.5 parsec 1 parsec=3.086×1016m 0.5 parsec=0.5×3.086×1016m=1.543×1016m

  4. 10 light years 1 light year=9.461×1015m 10 light years=10×9.461×1015m=9.461×1016m

From the above:

  • 108 angstrom = 0.01 m
  • 106 fermi = 10−9 m
  • 0.5 parsec = 1.543×1016 m
  • 10 light years = 9.461×1016 m

Clearly, 10 light years is the largest.

Step-By-Step Solution

Step 1

Convert 108 angstrom to meters: 108×10−10=10−2m

Step 2

Convert 106 fermi to meters: 106×10−15=10−9m

Step 3

Calculate 0.5 parsec in meters: 0.5×3.086×1016=1.543×1016m

Step 4

Calculate 10 light years in meters: 10×9.461×1015=9.461×1016m

Step 5

Compare values: 9.461×1016>1.543×1016>10−2>10−9

Final Answer

Option 4: 10 light year is the largest length