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

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