Numerical Problems

CHAPTER 01

Question 1.1

Speed of light, v = 3.0 x 10⁸ ms^-1

Travel time, t = 1 year

                         = 365 x 24 x 60 x 60 sec

                         = 31536000 sec

Since

            S = v . t

Therefore

            S = 3.0 x 10⁸ x 31536000

S = 9.5 x 10¹⁵ m                     Answer

Question 1.2

Question 1.3

Length, L = 15.3 cm

Width, W = 12.80 cm

Volume, V = ?

Since

Volume = Length x Width

V = L x W

V = 15.3 x 12.80

V = 196 cm²               Answer

Question 1.4

Question 1.5

Here  

Length of simple pendulum, l = 100 cm

Time for 20 vibrations = 40.2 s

Meter scale accuracy = 1 mm

Stop watch accuracy = 0.1 s

Now

l = 100 cm

  = 100/100  m

  = 1 m

T = 40.2/20

   = 2.01 s

FOR LENGTH

            Absolute uncertainty = 1 mm = 0.1 cm

            %age uncertainty = (0.1/100)  x  (100/100) = 0.1 %

FOR TIME

            Absolute uncertainty = 0.1 sec

Average uncertainty = 0.1/20 = 0.005 sce

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

Total uncertainty = 2 x 0.025 + 0.1 = 0.6 %

Total uncertainty for g = 9.76 x 0.6 /100 = 0.06

Thus g = 9.76 ± 0.06 ms^-2

Question 1.6

Question 1.7

v_f = v_i + a t

[LT^-1] = [LT^-1] + [LT^-2 T]

[LT^-1] = [LT^-1] + [LT^-1]

[LT^-1] = [LT^-1]

LHS = RHS                           Hence Proved

Question 1.8

According to question

            v ; ρ^a E^b

            v = constant . ρ^a E^b              - Eq (1)

For ρ

            ρ = m/v = [ML^-3]                 - Eq (2)

For E

            E = stress/strain

            E = (F/A)/(∆l/l) = [MLT^-2/L²] / [L/L] = [ML^-1T^-2]                   - Eq (3)

For v

            v = [LT^-1]                               - Eq (4)

Putting the value of ρ, E and v from Eq (2), (3) and (4) in Eq (1), we get

            v = constant . ρ^a E^b 

            [LT^-1] = constant [ML^-3]^a [ML^-1T^-2]^b

            [LT^-1] = constant M^a+b L^-3a-b T^-2b

Equating powers of corresponding quantities on both sides

a + b =0; - 3a - b = 1; -2b = -1

By solving, we get,

            a = -½, b = ½

Therefore from Eq (1)

            v = constant . ρ^a E^b

            v = constant . ρ^-½ E^½

 Question 1.9

            E = m C²

            mgh = m v²

            [M LT^-2 L] = [M (L T^-1)²]

            [M L² T^-2] = [M L² T^-2]

LHS = RHS                           Hence Proved

Question 1.10

            a ; rⁿ v^m

            a = constt. rⁿ v^m

            [L T^-2] = constt. [L]ⁿ [L T^-1]^m

            L T^-2 = constt. L^n+m  T^-m

Equating powers of corresponding quantities on both sides

n + m = 1; -m = -2

Þ        m = 2; n = -1                         Answer