If g is the acceleration due to gravity on the earth’s surface, the gain in the potential energy of object of mass m raised from the surface of the earth to a height equal to the radius R of the earth is ?

To find the gain in potential energy of an object of mass \( m \) when raised from the surface of the Earth to a height equal to the Earth's radius \( R \), we can use the formula for gravitational potential energy. The potential energy \( U \) at a height \( h \) in a gravitational field is given by the equation:

Gravitational Potential Energy Formula

The formula for gravitational potential energy is:

U = mgh

Where:

• U is the gravitational potential energy
• m is the mass of the object
• g is the acceleration due to gravity
• h is the height above the reference point

Considerations for Height Equal to Earth's Radius

When the object is raised to a height equal to the radius of the Earth, the height \( h \) is equal to \( R \). However, since the gravitational force decreases with distance from the center of the Earth, we need to adjust our calculations. The gravitational acceleration \( g \) varies with height, and at a distance \( R \) (which is the radius of the Earth), the gravitational acceleration is:

g' = g \cdot \frac{R}{R + R} = \frac{g}{2}

This means that at a height \( R \), the effective gravitational acceleration is half of the gravitational acceleration at the Earth's surface.

Calculation of Change in Potential Energy

Next, the change in potential energy when raising the object from the Earth's surface to a height \( R \) can be calculated using the potential energy difference:

U = mgh

At the surface of the Earth (height \( h = 0 \)):

U_1 = mg(0) = 0

At height \( h = R \):

U_2 = mg'R = m \cdot \frac{g}{2} \cdot R

Final Expression for Gain in Potential Energy

The gain in potential energy \( \Delta U \) is the difference between the potential energy at height \( R \) and at the surface:

ΔU = U_2 - U_1 = \left( m \cdot \frac{g}{2} \cdot R \right) - 0 = m \cdot \frac{g}{2} \cdot R

Summary

Thus, the gain in potential energy of an object of mass \( m \) raised from the surface of the Earth to a height equal to the radius of the Earth is:

ΔU = \frac{mgR}{2}

This expression illustrates how gravitational forces change with distance and how potential energy is significantly influenced by these factors. Understanding these principles is crucial in fields like physics, engineering, and even astrophysics, where gravitational interactions play a critical role.