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 × $10^{-8}$ $(\text{cm})^3\text s^{-2} \text g^{-1}$