Question

The gravitational potential in a region is given by V = 20 N kg⁻¹ (x + 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$\overrightarrow{i},\overrightarrow{j},\overrightarrow{k}$. (c) Calculate the magnitude of the gravitational force on a particle of mass 500 g placed at the origin.

Answer 1

(a) $V=\left(\frac{20\mathrm{N}}{\mathrm{kg}}\right)\left(x+y\right)$
$$\begin{gathered} \left[\frac{\mathrm{GM}}{\mathrm{R}}\right]=\left[\frac{{\mathrm{MLT}}^{-2}}{\mathrm{M}}\right]\left[\mathrm{L}\right] \\ \Rightarrow \left[\frac{{\mathrm{M}}^{-1}{\mathrm{L}}^3{\mathrm{T}}^{-2}{\mathrm{M}}^1}{\mathrm{L}}\right]=\left[\frac{{\mathrm{ML}}^2{\mathrm{T}}^{-2}}{\mathrm{M}}\right] \\ \Rightarrow \left[{\mathrm{M}}^0{\mathrm{L}}^2{\mathrm{T}}^{-2}\right]=\left[{\mathrm{M}}^0{\mathrm{L}}^2{\mathrm{T}}^{-2}\right] \\ \therefore \mathrm{LHS}=\mathrm{RHS} \end{gathered}$$

(b) The gravitational field at the point (x, y) is given by ${\overrightarrow{\mathrm{E}}}_{\left(x,y\right)}=-20\left(\frac{\mathrm{N}}{\mathrm{kg}}\right)\hat{i}-\left(\frac{20\mathrm{N}}{\mathrm{kg}}\right)\hat{j}$.

(c) $\overrightarrow{\mathrm{F}}=\overrightarrow{\mathrm{E}}m$
$$\begin{gathered} =0.5\mathrm{kg}\left[-\left(\frac{20\mathrm{N}}{\mathrm{kg}}\right)\hat{i}-\left(\frac{20\mathrm{N}}{\mathrm{kg}}\right)\hat{j}\right] \\ =-\left(10\mathrm{N}\right)\hat{i}-\left(10\mathrm{N}\right)\hat{j} \end{gathered}$$
$$\begin{gathered} \therefore \left|\overrightarrow{\mathrm{F}}\right|=\sqrt{\left(100\right)+\left(100\right)} \\ =10\sqrt{2}\mathrm{N} \end{gathered}$$