Question

Fill in the blanks by suitable conversion of units  

1. 1 \(\text {kg}\) \(\text m^{2}\text s^{-2}\) = _______ \(\text g\) \(\text{cm}^2\text s^{-2}\) .
2. 1 \(\text m\) = ______ly.
3. 3.0 \(\text m^{2}\text s^{-2}\) = ________ \(\text {km}\) \(\text h^{-2}\)
4. G = 6.67 × \(10^{-11}\) N \(\text m^2\) \((\text{kg})^{-2}\) =_______ \((\text{cm})^3\text s^{-2} \text g^{-1}\).

Answer 1

(a) Conversion of kg m² s⁻² to g cm² s⁻²

1 kg = 10³ g
1 m = 10² cm ⇒ 1 m² = 10⁴ cm²

Therefore:

1 kg m² s⁻² = (10³ g)(10⁴ cm²) s⁻²

= 10⁷ g cm² s⁻²

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(b) Conversion of metre to light-year

1 light-year (ly) ≈ 9.46 × 10¹⁵ m

Therefore:

1 m = 1 / (9.46 × 10¹⁵) ly

1 m ≈ 1.06 × 10⁻¹⁶ ly

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(c) Conversion of m² s⁻² to km h⁻²

1 m = 10⁻³ km ⇒ 1 m² = 10⁻⁶ km²

1 s = 1/3600 h ⇒ 1 s⁻² = 3600² h⁻²

So:

1 m² s⁻² = 10⁻⁶ × (3600)² km² h⁻²

= 12.96 km² h⁻²

Therefore:

3.0 m² s⁻² = 3.0 × 12.96

= 38.9 km² h⁻²

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(d) Conversion of gravitational constant G

Given:

G = 6.67 × 10⁻¹¹ N m² kg⁻²

We know:

1 N = 1 kg m s⁻²

So:

G = 6.67 × 10⁻¹¹ m³ kg⁻¹ s⁻²

Convert units:

• 1 m = 10² cm ⇒ 1 m³ = 10⁶ cm³
• 1 kg = 10³ g ⇒ kg⁻¹ = 10⁻³ g⁻¹

Therefore:

G = 6.67 × 10⁻¹¹ × 10⁶ × 10⁻³ cm³ s⁻² g⁻¹

G = 6.67 × 10⁻⁸ cm³ s⁻² g⁻¹

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Final Answers:

• (a) 10⁷ g cm² s⁻²
• (b) 1.06 × 10⁻¹⁶ ly
• (c) 38.9 km² h⁻²
• (d) 6.67 × 10⁻⁸ cm³ s⁻² g⁻¹