A planet of mass `M` has uniform density in a spherical volume of radius `R`. Calculate the work done by the external agent to deassemble the planet in eight identical spherical part against gravitational pull amongst its constitute particle.

Answer

To solve the problem of calculating the work done by an external agent to disassemble a planet of mass \( M \) and radius \( R \) into eight identical spherical parts against gravitational pull, we can follow these steps:

### Step 1: Determine the mass of each spherical part
Since the planet is divided into 8 identical parts, the mass of each part \( m \) can be calculated as:
\[ m = \frac{M}{8} \]

### Step 2: Determine the radius of each spherical part
The volume of the original planet is given by:
\[ V = \frac{4}{3} \pi R^3 \]
The volume of each smaller sphere must also equal the total volume divided by 8:
\[ \text{Volume of each part} = \frac{V}{8} = \frac{1}{8} \cdot \frac{4}{3} \pi R^3 = \frac{1}{6} \pi R^3 \]
Let \( r \) be the radius of each smaller sphere. Then:
\[ \frac{4}{3} \pi r^3 = \frac{1}{6} \pi R^3 \]
Cancelling \( \pi \) and solving for \( r \):
\[ r^3 = \frac{R^3}{8} \implies r = \frac{R}{2} \]

### Step 3: Calculate the gravitational potential energy of the original planet
The gravitational potential energy \( U \) of a solid sphere is given by:
\[ U = -\frac{3}{5} \frac{GM^2}{R} \]

### Step 4: Calculate the gravitational potential energy of the eight smaller spheres
The gravitational potential energy of the eight smaller spheres can be calculated as follows:
\[ U_f = 8 \left(-\frac{3}{5} \frac{G m^2}{r}\right) \]
Substituting \( m = \frac{M}{8} \) and \( r = \frac{R}{2} \):
\[ U_f = 8 \left(-\frac{3}{5} \frac{G \left(\frac{M}{8}\right)^2}{\frac{R}{2}}\right) \]
Calculating this gives:
\[ U_f = 8 \left(-\frac{3}{5} \frac{G \frac{M^2}{64}}{\frac{R}{2}}\right) = 8 \left(-\frac{3GM^2}{320R}\right) = -\frac{24GM^2}{320R} = -\frac{3GM^2}{40R} \]

### Step 5: Calculate the work done by the external agent
The work done \( W \) by the external agent is the change in gravitational potential energy:
\[ W = U_f - U_i \]
Substituting \( U_i \) and \( U_f \):
\[ W = \left(-\frac{3GM^2}{40R}\right) - \left(-\frac{3GM^2}{5R}\right) \]
Finding a common denominator (40R):
\[ W = -\frac{3GM^2}{40R} + \frac{24GM^2}{40R} = \frac{21GM^2}{40R} \]

### Final Answer
Thus, the work done by the external agent to disassemble the planet into eight identical spherical parts is:
\[ W = \frac{21GM^2}{40R} \]