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प्रश्न
The gravitational field in a region is given by
\[\vec{E} = \left( 5 N {kg}^{- 1} \right) \vec{i} + \left( 12 N {kg}^{- 1} \right) \vec{j}\] . (a) Find the magnitude of the gravitational force acting on a particle of mass 2 kg placed at the origin. (b) Find the potential at the points (12 m, 0) and (0, 5 m) if the potential at the origin is taken to be zero. (c) Find the change in gravitational potential energy if a particle of mass 2 kg is taken from the origin to the point (12 m, 5 m). (d) Find the change in potential energy if the particle is taken from (12 m, 0) to (0, 5 m).
उत्तर
Gravitational field,
\[\overrightarrow{E} = \left( 5N/kg \right) \hat i + \left( 12 N/kg \right) \hat j\]
(a)\[\overrightarrow{F} = m \vec{E}\]
\[= 2 kg \left[ \left( 5N/kg \right) \hat i + \left( 12 N/Kg \right) \hat j \right]\]
\[ = \left( 10 N \right) \hat i + \left( 24 N \right) \hat j \]
\[ \therefore \left| \overrightarrow{F} \right| = \sqrt{100 + 576} = \sqrt{676} = 26 N\]
(b)\[V = - \vec{E .} \vec{r}\]
Potential at (12 m, 0) \[= - 60 J/Kg\]
Potential at (0, 5 m) = \[- 60 J/kg\]
(c)
change in potential=final potential -initial potential
initial potential=potential at the origin=0
final potential=potential at (12,5)
\[V = - \overrightarrow{E} . \vec{r} \]
\[ = - (10 \hat i + 24 \stackrel\frown j ) . (12 \stackrel\frown i + 5 \stackrel\frown j)\]
\[ = - (120 + 120) J\]
\[ = - 240 J\]
(d)
\[∆ V = - \overrightarrow{E} . ∆ \overrightarrow{r}\]
\[ ∆ \vec{r} = (12 \stackrel\frown i+ 0 \stackrel\frown j) - (0 \stackrel\frown i+ 5 \stackrel\frown j )\]
\[ = 12 \stackrel\frown i - 5 \stackrel\frown j \]
\[ ∆ V = - (10 \stackrel\frown i + 12 \stackrel\frown j ) . (12 \stackrel\frown i - 5 \stackrel\frown j )\]
\[ = 0 J\]
