Gravitational Potential Energy

$-\frac{dV}{dr}=-\frac{k}{r^2}\Rightarrow \underset{10}{\overset{V}{\int }}dV=\underset{2}{\overset{3}{\int }}\frac{k}{r^2}dr$
$V-10=k\left[\frac{1}{2}-\frac{1}{3}\right]$
$V-10=\frac{k}{6}\Rightarrow V=11\text{volts}$

The given equation is:

where is a dimensionless constant, is the speed of light ( ), is the gravitational constant ( ), is Planck's constant ( ).

The dimensions of mass are:

The dimensions of each term in the equation are:

Substituting the dimensions into the equation:

Combine the dimensions of each term:

Equating powers of each dimension:

For mass :

For length :

Simplify:

For time :

Simplify:

Therefore, the value of is:

The work-energy theorem states that the minimum kinetic energy required for a particle to escape Earth's gravitational field is equal to the negative of the gravitational potential energy at that point. The gravitational potential energy at a distance from the center of Earth is given by: where: - is the gravitational constant, - is the mass of Earth, - is the mass of the particle, and - is the distance from the center of the Earth. For the given question, the particle is at a distance from the center of the Earth, where is the radius of Earth. Therefore, the potential energy at this point is: The minimum kinetic energy required for the particle to leave Earth's field is the amount of energy needed to overcome this gravitational potential energy. Thus, the minimum kinetic energy is: Now, using the relation (where is the acceleration due to gravity at Earth's surface), we can rewrite the expression for kinetic energy as: Thus, the minimum kinetic energy required for the particle to escape Earth's field is . Therefore, the correct answer is (3) .

Step 1: Expression for gravitational potential energy.
For three masses placed at the vertices of an equilateral triangle of side , total gravitational potential energy is:
Step 2: Calculating initial potential energy.
Step 3: Calculating final potential energy.
Step 4: Work done in rearrangement.
Step 5: Final conclusion.
The work done in rearranging the masses is .