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 \[\vec{i} , \vec{j} , \vec{k}\] . (c) Calculate the magnitude of the gravitational force on a particle of mass 500 g placed at the origin.

Answer 1

(a)\[V = \left( \frac{20 N}{kg} \right) \left( x + y \right)\] 

\[{\left[ \frac{GM}{R} \right]} = {\left[ \frac{{MLT}^{- 2}}{M} \right]} \left[ L \right]\]

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

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

\[ \therefore LHS = RHS\]

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

(c)\[\overrightarrow{F} = \overrightarrow{E} m\]

\[= 0 . 5 kg \left[ - \left( \frac{20 N}{kg} \right) \hat i - \left( \frac{20 N}{kg} \right) \hat j \right]\]

\[ = - \left( 10 N \right) \hat i - \left( 10 N \right) \hat j\]

\[\therefore \left| \overrightarrow{F} \right| = \sqrt{\left( 100 \right) + \left( 100 \right)}\]

\[ = 10\sqrt{2} N\]