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Chapter 1: Problem 59

If value of gravitational constant in MKS is $6.67 \times 10^{-11} \mathrm{Nm}^{2} / \mathrm{kg}^{2}\( then value of \)\mathrm{G}$ in \(\mathrm{CGS}=\ldots \ldots \ldots \ldots . .\left[\left(\right.\right.\) dyne \(\left.\left.-\mathrm{cm}^{2}\right) /\left(\mathrm{gm}^{2}\right)\right]\) (a) \(6.67 \times 10^{-9}\) (b) \(6.67 \times 10^{-7}\) (c) \(6.67 \times 10^{-8}\) (d) \(6.67 \times 10^{-5}\)

Short Answer

Expert verified
The value of G in the CGS system is \(6.67 \times 10^{-10} \frac{\mathrm{dyne} \cdot \mathrm{cm}^2}{\mathrm{gm}^2}\). However, none of the given options match the calculated value.

Step by step solution

01

Determine the conversion factors between the systems

: In the given exercise, we have: 1. Length: MKS (meters) to CGS (centimeters): 1 meter = 100 centimeters 2. Mass: MKS (kilograms) to CGS (grams): 1 kilogram = 1000 grams 3. Force: MKS (Newtons) to CGS (dynes): 1 Newton = 100,000 dynes
02

Apply the conversion factors

: The value of G in MKS is given as \(6.67 \times 10^{-11} \mathrm{Nm}^{2} / \mathrm{kg}^{2}\). To convert this into CGS: 1. Replace meters with centimeters: \(\mathrm{Nm}^{2} \rightarrow 100 \times 100 \mathrm{cm}^{2}\) 2. Replace kilograms with grams: \(/ \mathrm{kg}^{2} \rightarrow / (1000 \times 1000 \mathrm{gm}^{2})\) 3. Replace Newtons with dynes: \(N \rightarrow 100,000 \mathrm{dyne}\) Now apply all these conversions simultaneously: \(6.67 \times 10^{-11} \times (\frac{100,000 \cdot 100 \cdot 100}{1000 \cdot 1000}) \frac{\mathrm{dyne} \cdot \mathrm{cm}^2}{\mathrm{gm}^2}\)
03

Simplify and compare with the given options

: Simplify the expression: \(6.67 \times 10^{-11} \times (\frac{10,000,000}{1,000,000}) \frac{\mathrm{dyne} \cdot \mathrm{cm}^2}{\mathrm{gm}^2} = 6.67 \times 10^{-11} \times 10 \frac{\mathrm{dyne} \cdot \mathrm{cm}^2}{\mathrm{gm}^2} = 6.67 \times 10^{-10} \frac{\mathrm{dyne} \cdot \mathrm{cm}^2}{\mathrm{gm}^2}\) Compare the calculated value with the given options: (a) \(6.67 \times 10^{-9}\) (b) \(6.67 \times 10^{-7}\) (c) \(6.67 \times 10^{-8}\) (d) \(6.67 \times 10^{-5}\) None of the given options match the calculated value: \(6.67 \times 10^{-10} \, \mathrm{dyne} \cdot \mathrm{cm}^2 / \mathrm{gm}^2\).

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