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Step-by-Step Solution
Step 1: Identify the Relevant Concept
The problem deals with gravitational potential energy changes for a body when raised to a certain height above the Earth's surface. Specifically, we use the formula for gravitational potential energy in a two-body system (Earth and the object) and apply it at the surface and at a height equal to the Earth’s radius.
Step 2: Write the General Expression for Gravitational Potential Energy
For a body of mass $m$ at a distance $r$ from the center of the Earth (mass $M$), the gravitational potential energy $U$ is given by:
$ U = -\frac{GMm}{r} $
where:
$G$ is the universal gravitational constant.
$M$ is the mass of the Earth.
Step 3: Calculate Potential Energy on the Surface of the Earth
On the surface of the Earth, $r = R$ (where $R$ is Earth’s radius). So the potential energy is:
$ U_{\text{surface}} = -\frac{GMm}{R} $
Step 4: Calculate Potential Energy at Height $R$ Above the Surface
When the object is at a height $h = R$ above the surface, its distance from the Earth’s center becomes $r = R + R = 2R$. Hence, the potential energy at that position is:
$ U_{\text{height}} = -\frac{GMm}{2R} $
Step 5: Determine the Increase in Potential Energy
The gain in potential energy $\Delta U$ is the difference between the final potential energy and the initial potential energy:
$ \Delta U = U_{\text{height}} - U_{\text{surface}} $
Substituting the expressions:
$ \Delta U = \left(-\frac{GMm}{2R}\right) - \left(-\frac{GMm}{R}\right)
= -\frac{GMm}{2R} + \frac{GMm}{R}
= \frac{GMm}{2R}
$
Step 6: Relate $GM/R^2$ to $g$
The acceleration due to gravity at the Earth’s surface is given by $g = \frac{GM}{R^2}$. Multiplying both sides by $R$ gives:
$ \frac{GM}{R} = gR
$
This allows us to express the gain in potential energy in terms of $m$, $g$, and $R$:
$ \Delta U = \frac{GMm}{2R}
= \frac{m(gR)}{2}
= \frac{1}{2} m g R
Step 7: Write the Final Answer
Thus, the gain in potential energy when the object is raised from the surface of the Earth to a height equal to the Earth’s radius is:
$ \Delta U = \frac{1}{2} \, m \, g \, R
$
