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) 1 \(\text {kg}\) \(\text m^{2}\text s^{-2}\) = \(\underline{10^7 g \;\text {cm}^2 \text s^{-2}}.\)
Explanation:
1 kg = \(10^3\) g
1 \(\text m^2\) = \(10^4 \; \text {cm}^2\)
1 kg \(\text m^{2}\text s^{-2}\) = 1 kg × 1\(\text m^2\) × 1 \(\text s^{-2}\)
=\(10^3\) g × \(10^4 \; \text {cm}^2\) × 1 \(\text s^{-2}\) = \(10^7 g \;\text {cm}^2 \text s^{-2}\)

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(b) 1 \(\text m\) = \(\underline{1.057\times10^{-16}\text{ly}}.\)
Explanation:
Light year is the total distance travelled by light in one year.
1 ly = Speed of light × One year 
= (3 × \(10^8\)m/s) × (365 × 24 × 60 × 60 s) 
= 9.46 × \(10^{15}\)m 
∴ 1 m = \(\frac{1}{9.46 \times 10^{15}}\) = 1.057 × \(10^{-16}\) ly

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(c) 3.0 \(\text m^{2}\text s^{-2}\) = \(\underline{3.88\times10^{-4}\text{km}\text h^{-2}}.\)
Explanation:
1 m = \(10^{-3}\) km
Again , 1 s = \(\frac{1}{3600}\) h
1 \(\text s^{-1}\) = 3600 \(\text h^{-1}\)
1 \(\text s^{-2}\)=\((3600)^2 \text h^{-2}\)
∴ 3 m \(\text s^{-2}\)= (3 × \(10^{-3}\) km) × (\((3600)^2 \text h^{-2}\)) = 3.88 × \(10^{-4}\) km \(\text h^{-2}\)

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(d) G = 6.67 × \(10^{-11}\) N \(\text m^2\) \((\text{kg})^{-2}\) = \(\underline{6.67\times 10^{-8}(\text{cm})^3\text s^{-2} \text g^{-1}}.\)
Explanation:
1 N = 1 kg m\(\text s^{-2}\) 
1 kg = \(10^{-3}\) \(\text g^{-1}\) 
1 \(\text m^3\) = \(10^6 \text {cm}^3\)
∴ 6.67 ×\(10^{-11} \text N \text m^2 \text {kg}^{-2}\)= 6.67 ×\(10^{-11}\) × (1 kg m\(\text s^{-2}\)) (1 \(\text m^2\) ) (1 \(\text s^{-2}\))
= 6.67 × \(10^{-11}\) × (1 kg × 1 \(\text m^3\) \(\times 1 \text s^{-2})\)
= 6.67 × \(10^{-11}\) × (\(10^{-3}\;\text g^{-1}\)) × \((10^6\; \text {cm}^3) \times (1 \text s^{-2})\)
= 6.67 × \(10^{-8}\) \((\text{cm})^3\text s^{-2} \text g^{-1}\)