A particle of mass 10 g is kept on the surface of a uniform sphere of mass 100 kg and radius 10 cm.

Q: Why is the initial potential energy negative?

Gravitational force is always attractive. This means the system does work to bring masses together from infinity, resulting in a negative potential energy relative to the zero at infinity.

Q: Why do we take the final potential energy as zero?

As the distance between the masses approaches infinity, the gravitational force between them approaches zero. By convention, we set the potential energy at this state of zero interaction to be zero.

Q: Do we need to consider the radius of the small particle?

No, the particle is treated as a point mass because its size is negligible compared to the radius of the large sphere.

Q: What if the particle was moved to a finite distance instead of infinity?

You would simply calculate the final potential energy at that specific distance $r$ using $-GMm/r$ and subtract the initial potential energy from it.

Q: Is this work done related to escape velocity?

Yes, absolutely! If you were to provide this exact amount of work as initial kinetic energy, the particle would have just enough speed to escape the sphere's gravitational pull forever.

Question

A particle of mass $10 	ext{ g}$ is kept on the surface of a uniform sphere of mass $100 	ext{ kg}$ and radius $10 	ext{ cm}$. Find the work to be done against the gravitational force between them, to take the particle far away from the sphere, (you may take $G = 6.67 	imes 10^{-11} 	ext{ Nm}^2/	ext{kg}^2$)

Options:

(A) $13.34 	imes 10^{-10} 	ext{ J}$

(B) $3.33 	imes 10^{-10} 	ext{ J}$

(C) $6.67 	imes 10^{-9} 	ext{ J}$

(D) $6.67 	imes 10^{-10} 	ext{ J}$

Sigma Percentile · Accepted Answer

### The Setup: A Tiny Particle on a Massive Sphere

Imagine you are standing next to a massive, uniform sphere weighing $100 	ext{ kg}$. Resting gently on its surface is a tiny particle with a mass of just $10 	ext{ g}$. Our mission is to find out exactly how much work we need to do to take this tiny particle and move it infinitely far away from the sphere.

This is a classic problem of overcoming gravitational attraction. Gravity is constantly pulling the particle towards the center of the sphere, so we have to put in effort—or **work**—to pull them apart.

### The Master Equation: Work and Potential Energy

In physics, when we deal with conservative forces like gravity, the work done by an external agent to move an object slowly from one point to another is exactly equal to the change in the system's potential energy.

Mathematically, we write this as:

$$W = \Delta U = U_f - U_i$$

Here, $U_f$ is the final potential energy when the particle is at infinity, and $U_i$ is the initial potential energy when the particle is on the surface of the sphere.

### Calculating the Potential Energies

Let's figure out these two energy states.

First, what happens at infinity? As the distance between the two masses becomes infinitely large, the gravitational pull between them drops to zero. By convention, we define the gravitational potential energy at an infinite distance to be zero.

$$U_f = 0$$

Now, what about the initial state? When the particle is on the surface of the sphere, it is at a distance $R$ from the sphere's center. The formula for gravitational potential energy is:

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

The negative sign is crucial here! It tells us that the system is bound together. We have to supply energy to break this bond.

### The Final Calculation

Substituting our potential energies back into the work equation, we get:

$$W = 0 - \left(-\frac{GMm}{R}ight) = \frac{GMm}{R}$$

This makes perfect sense: the work done is positive because we are adding energy to the system. Now, before we plug in the numbers, we must ensure all our units are in standard SI format to avoid any silly mistakes.

- Mass of the sphere, $M = 100 	ext{ kg}$
- Mass of the particle, $m = 10 	ext{ g} = 10 	imes 10^{-3} 	ext{ kg}$
- Radius of the sphere, $R = 10 	ext{ cm} = 0.1 	ext{ m}$
- Gravitational constant, $G = 6.67 	imes 10^{-11} 	ext{ Nm}^2/	ext{kg}^2$

Let's substitute these values into our simplified work equation:

$$W = \frac{6.67 	imes 10^{-11} 	imes 100 	imes (10 	imes 10^{-3})}{0.1}$$

Let's group the powers of 10 to make the calculation smoother:

$$W = \frac{6.67 	imes 10^{-11} 	imes 10^2 	imes 10^{-2}}{10^{-1}}$$

The $10^2$ and $10^{-2}$ in the numerator cancel each other out perfectly. We are left with:

$$W = \frac{6.67 	imes 10^{-11}}{10^{-1}}$$

$$W = 6.67 	imes 10^{-10} 	ext{ J}$$

And there we have it! The work required to move the particle to infinity is a minuscule $6.67 	imes 10^{-10} 	ext{ Joules}$.

**A Quick Thought Experiment:** What if, instead of carrying the particle slowly, we just launched it from the surface? The initial kinetic energy we would need to give it would be exactly equal to this work done. This beautiful equivalence is the fundamental principle behind calculating **escape velocity**!

Select Answer:

Visualized Solution

$W = ?$

$W = Δ U$

$U_{i} and U_{f}$

$W = \frac{GM m}{R}$

$Substitution$

$W = 6.67 \times 1 0^{- 10} J$

$Escape Energy$

The Sigma Insight: Gravitational Potential and Potential Energy

The Setup

The Master Equation

Calculating the Potential Energies

The Final Calculation

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Select Answer:

Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } W=?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } W=ΔU

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Ui​ and Uf​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } W=RGMm​

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Substitution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } W=6.67×10−10 J

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Escape Energy