Energy required to move a body of mass m from an orbit of radius 2R to 3R is
(a) GMm / 12R^2
(b) GMm

Q: Why didn't we use the total energy formula $E = -GMm/2r$ since the word 'orbit' is used?

Strictly speaking, moving between orbits requires changing kinetic energy too. However, $\Delta E = GMm/12R$. Option (a) is $GMm/12R^2$, which has incorrect dimensions for energy (it's a force). Thus, the question intends for you to only calculate the work done against gravity (change in potential energy).

Q: Why is potential energy negative?

Potential energy is defined to be zero at infinity. Since gravity is an attractive force, work is done *by* the gravitational field as a mass approaches from infinity, making the potential energy negative at any finite distance.

Q: Does the path taken from $2R$ to $3R$ matter?

No. Gravitational force is a conservative force, meaning the work done depends only on the initial and final positions, not the path taken.

Q: What if the question asked for the change in kinetic energy?

In a circular orbit, kinetic energy is $K = GMm/2r$. The change would be $\Delta K = GMm/2(3R) - GMm/2(2R) = -GMm/12R$.

Q: How do we know $R$ is the distance from the center and not the surface?

In orbital mechanics, unless specified as 'height above the surface', the radius of an orbit is always measured from the center of the central body.

Question

Energy required to move a body of mass $m$ from an orbit of radius $2R$ to $3R$ is

Options:

(A) $\frac{GMm}{12R^2}$

(B) $\frac{GMm}{3R^2}$

(C) $\frac{GMm}{8R}$

(D) $\frac{GMm}{6R}$

Sigma Percentile · Accepted Answer

### Visualizing the Setup

Imagine a planet of mass $M$ resting at the center of our coordinate system. A smaller body of mass $m$ is initially located at a distance of $2R$ from the center of the planet. Our objective is to find the energy required to move this body to a farther distance of $3R$.

### The "Orbit" Trap: Total vs Potential Energy

There is a subtle trap in this question. The problem uses the word **"orbit"**. Normally, when a body moves from one circular orbit to another, we must account for the change in its kinetic energy as well as its potential energy. The total energy of a body in a circular orbit of radius $r$ is given by $E = -\frac{GMm}{2r}$.

If we were to calculate the change in total energy, it would be:

$$\Delta E = \left(-\frac{GMm}{2(3R)}\right) - \left(-\frac{GMm}{2(2R)}\right) = \frac{GMm}{12R}$$

However, if you look closely at the given options, $\frac{GMm}{12R}$ is not present. Option (a) is $\frac{GMm}{12R^2}$, which has an $R^2$ in the denominator. This makes it dimensionally incorrect for energy (it has the dimensions of force). This dimensional mismatch is a massive hint! It tells us that the question is not asking for the change in total orbital energy, but rather just the **work done against the gravitational force**, which is simply the change in gravitational potential energy.

### Calculating the Energy Difference

The gravitational potential energy $U$ of a two-body system separated by a distance $r$ is given by:

$$U = -\frac{GMm}{r}$$

Let's write down the potential energies for the initial and final states.

Initially, the body is at a distance of $2R$:

$$U_i = -\frac{GMm}{2R}$$

Finally, the body is moved to a distance of $3R$:

$$U_f = -\frac{GMm}{3R}$$

### Final Calculation

The energy required to move the body is the difference between the final and initial potential energies:

$$\Delta U = U_f - U_i$$

Substituting our values into the equation:

$$\Delta U = \left(-\frac{GMm}{3R}\right) - \left(-\frac{GMm}{2R}\right)$$

$$\Delta U = \frac{GMm}{2R} - \frac{GMm}{3R}$$

Taking $\frac{GMm}{R}$ as a common factor:

$$\Delta U = \frac{GMm}{R} \left(\frac{1}{2} - \frac{1}{3}\right)$$

$$\Delta U = \frac{GMm}{R} \left(\frac{3 - 2}{6}\right)$$

$$\Delta U = \frac{GMm}{6R}$$

This perfectly matches option (d). By paying close attention to the dimensions of the options, we successfully navigated the ambiguity of the word "orbit" and arrived at the correct physical interpretation.

Animated Solution for Physics - Gravitation: Energy required to move a body of mass $m$ from an orbit of radius $2 R$ to $3 R$ is

Select Answer:

Visualized Solution

Visual Anchor

The Trap

Logic Bridge

Initial Potential Energy

Final Potential Energy

Energy Difference

Final Answer

The Sigma Insight: Gravitational Potential and Potential Energy

Visualizing the Setup

The "Orbit" Trap

Calculating the Energy Difference

Final Calculation

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Animated Solution for Physics - Gravitation: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Energy required to move a body of mass m from an orbit of radius 2R to 3R is