Question

The gravitational field in a region is given by $g = 5\dfrac{N}{kg}\widehat{i} + 12\dfrac{N}{kg}\widehat{j}$. The change in the gravitational potential energy of a particle of mass 2kg when it is taken from origin to a point \[\left( 7m,-3m \right)\].
$\begin{align}
  & A.71J \\
 & B.13\sqrt{58}J \\
 & C.-71J \\
 & D.-2J \\
\end{align}$

Answer 1

Hint: The potential at a certain point is given as,
$V=\int{gdx}$
And the location is,
\[\left( 7m,-3m \right)\]
Therefore V is the dot product of the gravitational field and distance between them.
Multiply this value of V with mass of the body. Thus the value of gravitational potential energy will be obtained.

Complete step by step answer:
Here in this question, it is given that gravitational field
$g=5\dfrac{N}{kg}\widehat{i}+12\dfrac{N}{kg}\widehat{j}$
Mass of the particle is given as 2kg.
The coordinates of the point is (7m, − 3m).
Therefore the distance between the point and the origin is$7\widehat{i}-3\widehat{j}$.
We know that potential at a given point is
$V=\int{gdx}$
Here E=g
Substituting the value of g and dx I the equation gives,
$V=\left( \left( 5\widehat{i}+12\widehat{j} \right)\cdot \left( 7\widehat{i}-3\widehat{j} \right) \right)$
After completing the dot product the equation of potential becomes
$V=35-36=-1$
Now we know that change in potential energy is
 \[U=\text{ }MV\]
Where M is the mass of the particle which is given as 2kg.
Therefore,\[U=2\times -1=-2J\]
Hence the correct answer is option D.

Additional information:
Gravitational energy is also a potential energy in which a physical object with mass has in relation to another massive object due to gravity. In short it is potential energy associated with the gravitational field.

Note:
Gravitational potential energy, or U, is energy that is stored due to an object's position or height above Earth's surface. Gravity is the force of attraction that attracts objects to the centre and causes things to fall to Earth. The most general expression for gravitational potential energy arises from the law of gravity. It is similar to the work done against gravity to bring a mass to a given point.