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Chapter 6: Problem 673

Correct form of gravitational law is (A) $\mathrm{F}=-\left[\left(\mathrm{Gm}_{1} \mathrm{~m}_{2}\right) / \mathrm{r}^{2}\right]$ (B) $\mathrm{F}^{-}=-\left[\left(\mathrm{Gm}_{1} \mathrm{~m}_{2}\right) / \mathrm{r}^{2}\right]$ (C) $\mathrm{F}^{-}=-\left[\left(\mathrm{Gm}_{1} \mathrm{~m}_{2}\right) / \mathrm{r}^{2}\right] \hat{\mathrm{r}}$ (B) $\mathrm{F}^{\rightarrow}=-\left[\left(\mathrm{Gm}_{1} \mathrm{~m}_{2}\right) / \mathrm{r}^{3}\right] \mathrm{r}^{-}$

Short Answer

Expert verified
The correct form of the gravitational law is given by Option C: \(F^{-} = -\frac{(Gm_1m_2)}{r^2}\hat{r}\), as it includes the gravitational constant, the masses of both objects, the distance between their centers, and the force vector \(\hat{r}\) pointing along the line connecting the centers of the objects.

Step by step solution

01

Option A

Option A states: \(F = -\frac{(Gm_1m_2)}{r^2}\) This option does include the gravitational constant and the masses of both objects, and the force goes inversely with the square of the distance between them. However, this option does not include any direction information.
02

Option B

Option B states: \(F^{-} = -\frac{(Gm_1m_2)}{r^2}\) This option is nearly the same as option A, but with a minus sign. The inclusion of a minus sign could indicate the attractive nature of the gravitational force, but this option still does not include any direction information.
03

Option C

Option C states: \(F^{-} = -\frac{(Gm_1m_2)}{r^2}\hat{r}\) This option includes the gravitational constant, the masses of both objects, the distance between their centers, and a unit vector in the radial direction, \(\hat{r}\), which indicates the direction of the force. This seems to be the correct form of the gravitational law.
04

Option D

Option D states: \(F^{\rightarrow} = -\frac{(Gm_1m_2)}{r^3}r^{-} \) This option includes the gravitational constant and the masses of both objects. However, the force is inversely proportional to the cube of the distance, not the square, which is incorrect. Additionally, this option does not have a clear direction. Based on the analysis of all the options, the correct form of the gravitational law is found in Option C.

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