Fill in the blanks by suitable conversion of units

Fill in the blanks by suitable conversion of units:
(a) 1 kg m²s^–2= ...g cm² s^–2
(b) 1 m =... ly
(c) 3.0 m s^–2=... km h^–2
(d) G= 6.67 × 10^–11 N m² (kg)^–2=... (cm)³s^–2 g^–1.

Solution

(a) 1 kg m²s^–2= ...g cm² s^–2

1 kg m² s^–2 = 1 kg × 1 m² × 1 s^–2
we know that

1 kg = 10³ g
1 m = 100 cm = 10² cm

Plug these values we get

1 kg × 1 m² × 1 s^–2 =10³ g × (10² cm)² × 1 s^–2 = 10³ g × 10⁴ cm² × 1 s^–2 = 10⁷ g cm² s^–2
(b) one light year is the distance travelled by sun light in 1 year

Use formula

Distance = speed × time

Speed of light = 3 × 10⁸ m/s

Time = 1 year = 365 days  = 365 × 24 hours  = 365 × 24 × 60 × 60  sec

Plug these values in above formula we get

1 light year distance =  (3 × 10⁸ m/s) × (365 × 24 × 60 × 60 s) = 9.46 × 10¹⁵ m

9.46 × 10¹⁵ m = 1 ly

So that 1 m = 1/ 9.46 × 10¹⁵ ly  = 1.06 × 10^-16 ly 
(c) 3.0 m s–2=... km h^–2
1 hour = 3600 sec  so that 1 sec = 1/3600 hour 

1 km = 1000 m        so that 1 m  = 1/1000 km

3.0 m s–2  = 3.0 (1/1000 km)( 1/3600 hour )^-2 = 3.0 × 10^–3 km × ((1/3600)^-2 h^–2)

= 3.0 × 10^–3 km × (3600)² h^–2 = 3.88 × 10⁴ km h^–2
Answer 3.0 m s–2=3.88 × 10⁴ km h^–2

(d) G= 6.67 × 10^–11 N m² (kg)^–2=... (cm)³s^–2 g^–1.

Given that

G= 6.67 × 10^–11 N m² (kg)^–2

We know that

 1 N = 1 kg m s^–2
1 kg = 10³ g
1 m = 100 cm = 10² cm 

plug above values, we get  

=> 6.67 × 10^–11 N m² kg^–2 = 6.67 × 10^–11 × (1 kg m s^–2) (1 m²) (1Kg^–2)

Solve and cancel out the units we get
=>  6.67 × 10^–11 × (1 kg ^–1 × 1 m³ × 1 s^–2)

Plug above values to convert Kg  to g and m to cm
=>  6.67 × 10^–11 × (10³ g)^-1 × (10² cm)³ × (1 s^–2)

=>  6.67 × 10^–11 × 10^-3 g^-1 × 10⁶ cm³ × (1 s^–2)
=>  6.67 × 10^–8 cm³ s^–2 g^–1