A body of mass \(\mathrm{m}\) is taken from earth surface to the height \(\mathrm{h}\) equal to radius of earth, the increase in potential energy will be (A) \(\operatorname{mg} R\) (B) \((1 / 2) \mathrm{mgR}\) (C) \(2 \mathrm{mg} \mathrm{R}\) (D) \((1 / 4) \mathrm{mgR}\)

The formula for gravitational potential energy (PE) is: PE = mgh At the initial point, the body is on the Earth's surface, so h = 0. Therefore, the initial potential energy is: PE_initial = mg(0) = 0

At the final point, the body is at a height h = R. To find the potential energy at this height, we need to consider the gravitational force acting on the body, which is given by the formula: F_gravity = G (m1 * m2) / r^2 where G is the gravitational constant, m1 is the mass of Earth, m2 is the mass of the body, and r is the distance between the centers of mass. At the final point, r is equal to the sum of Earth's radius and the height (R + R), so r = 2R. Replacing r with 2R in the formula for gravitational force, we get: F_gravity = G (m1 * m2) / (2R)^2 Now, we know that F_gravity = mg, so: mg = G (m1 * m2) / (2R)^2 By rearranging the formula, we can find the acceleration due to gravity, g, at the final point: g_final = G (m1) / (2R)^2 Then, we can use the formula for potential energy to find PE at the final point: PE_final = m * g_final * R PE_final = m * (G (m1) / (2R)^2) * R

The increase in potential energy is the difference between the potential energy at the final point and the initial point: Increase in PE = PE_final - PE_initial Since PE_initial is 0, we have: Increase in PE = PE_final Increase in PE = m * (G (m1) / (2R)^2) * R Now, comparing this expression with the given options, we can identify the correct answer as: (C) \(2 \mathrm{mg} \mathrm{R}\)