# Solve the following :

Question:

The gravitational potential in a region is given by $\mathrm{V}=(20 \mathrm{~N} / \mathrm{kg})(\mathrm{x}+\mathrm{y})$. (a) Show that the equation is dimensionally correct. (b) Find the gravitational field at the point $(x, y)$. Leave your answer in terms of the unit vectors $i, j, k$. (c) Calculate the magnitude of the gravitational force on a particle of mass 500

Solution:

$V=20 \frac{N}{k g}(x+y)$

Dimension of L.H.S $=[\mathrm{V}]$

$=\left[\frac{G M}{R}\right]$

$=\frac{\left[M^{-1} L^{3} T^{-2}\right][M]}{[L]}$

$=\left[M^{0} L^{2} T^{-2}\right]$

Dimension of RHS $=\frac{\left[M^{1} L^{1} T^{-2}\right]}{[M]}[L]=\left[M^{0} L^{2} T^{-2}\right]$

$\because L H S=R H S$

Hence, dimensionally correct.

(b) $\overrightarrow{\vec{E}}=-\left[\frac{\delta v}{d x} \hat{\imath}+\frac{\delta v}{d y} \hat{\jmath}\right]$

$\vec{E}=-\left[\frac{\delta}{d x}(20 x+20 y) \hat{\imath}+\frac{\delta}{d y}(20 x+20 y) \hat{\jmath}\right]$

$\vec{E}=-[20 \hat{\imath}+20 \hat{\jmath}]$

$\vec{E}=-20(\hat{\imath}+\hat{\jmath})$

(c) $\vec{F}=m \vec{E}$

$\vec{F}=(0.5)(-20 \hat{\imath}-20 \hat{\jmath})$

$\vec{F}=-10 \hat{\imath}-10 \hat{\jmath}$

$|\vec{F}|=\sqrt{(-10)^{2}+(-10)^{2}}=10 \sqrt{2}$