Understanding

Light Year: The distance travelled by light in one year is called light year.

As light year is a distance, so the unit of light year is metre (m).

How many seconds are there in one year?

1 year = 365 days
          = 365 x 24 hours
          = 365 x 24 x 60 minutes
          = 365 x 24 x 60 x 60 seconds
          = 31536000 seconds

Solution

Given that,
        speed of light = 3.0 x 10⁸ ms^-1
        time = 1 year = 31536000 s

We know that

then

         distance = speed of light x time

         distance = 3.0 x 10⁸ x 31536000

                     = 3.0 x 10⁸ x 3.1536 x 10⁷

                     = 9.5 x 10¹⁵ m

Hence, one light year has 9.5 x 10¹⁵ metres.

Understanding

nano (n) = 10^-9

       1 nanosecond (ns) = 10^-9 s

Solution

a)     1 year = 365 days
                  = 365 x 24 hours
                  = 365 x 24 x 60 minutes
                  = 365 x 24 x 60 x 60 seconds
                  = 31536000 secdons
                  = 3.1536 x 10⁷ secdons

Hence, 1 year = 3.1536 x 10⁷ s

b)     Consider,
                1 year = 3.1536 x 10⁷ seconds
                         = 3.1536 x 10⁷ x 10⁹ x 10^-9 seconds
                         = 3.1536 x 10⁷ x 10⁹ nanoseconds
                         = 3.1536 x 10¹⁶ nanoseconds

Hence, 1 year = 3.1536 x 10¹⁶ ns

c)     As     31536000 seconds = 1 year

         then

                                = 3.17 x 10^-8 years

Hence, 1 second = 3.17 x 10^-8 years

Understanding

Area of a rectangle = length x width

Solution

Given that,
        length of plate = 15.3 cm
        width of plate = 12.80 cm

We know that

     Area of rectangular plate = length of the plate x width of the plate 

                                        = 15.3 x 12.80

                                        = 195.84 cm²

                                        = 196 cm²

Hence, Area of the rectangular plate is 196 cm² .

Understanding

Significant Figures: In any measurement, the accurately known digits and the first doubtful digit are called significant figures.

Solution

Given numbers are,
                          2.189, 0.089, 11.8, 5.32

By adding, we have,

                          2.189 + 0.089 + 11.8 + 5.32 

                          = 19.398

Since the smallest number of decimal places is 1 in the given quantity 11.8, so the result is 19.4 kg.

Understanding

Time Period: It is the the time to complete one vibration or oscillation.

uncertainty in the time period of a vibrating body

Solution

Given data,
        Length of simple pendulum, l = 100 cm = 1 m
           with least count of metre scale = 1 mm = 0.001 m

        Time for 20 vibrations = 40.2 s
           with least count of stop watch = 0.1 s

Now consider,
        time for 20 vibrations = 40.2 s
        time for 1 vibration = 40.2 / 20 = 2.01 s

         Time Period, T = 2.01 s

        uncertainty in time period = 0.1 / 20 = 0.005 s
        time for 1 vibration = 40.2 / 20 = 2.01 s

Thus,
           T = 2.01 ± 0.005 s

    and   l = 1 ± 0.001 m 

Given formula is

$$T = 2\pi \sqrt{\frac{l}{g}} $$

By simlifying the above formula for 'g' we have

$$ \frac{T}{2\pi} = \sqrt{\frac{l}{g}} $$

$$ (\frac{T}{2\pi})^2 = \frac{l}{g} $$

$$ g = \frac{4 \pi ^ 2 l}{T ^ 2} $$

The %age uncertainty in l = (0.001/1) x (100/100) = 0.1%

The %age uncertainty in T = (0.005/2.01) x (100/100) = 0.25%

Total uncertainty in the value of g

                       = 0.1% + 2 x 0.25% = 0.1% + 0.5% = 0.6%

Now using the formula, value of 'g'

                  g  = 9.76 ms^-2

Then  g = 9.76 m/s²  with 0.6% uncertainty

   As         (0.6/100) x 9.76 = 0.06

Thus      g = (9.76 ± 0.06) ms^-2