Mass `M`, of a planet earth is uniformly distributed over a spherical volume of radius `R`. Calculate the energy needed to deassemble the planet against the gravitational pull amongst its consituent particles. Given
`mR = 2.5 xx 10^(31) kg m` and `g = 10 ms^(-2)`.

Explanation

To calculate the energy needed to disassemble the planet against the gravitational pull among its constituent particles, we can follow these steps:

### Step 1: Determine the Density of the Planet
The density \( \rho \) of the planet can be calculated using the formula for the mass of a sphere:

\[ \rho = \frac{M}{\frac{4}{3} \pi R^3} \]

Given that \( mR = 2.5 \times 10^{31} \, \text{kg m} \), we can express \( M \) as:

\[ M = \frac{2.5 \times 10^{31}}{R} \]

### Step 2: Consider a Spherical Shell
Consider a thin spherical shell of radius \( x \) and thickness \( dx \). The mass \( m_1 \) of this shell can be expressed as:

\[ m_1 = \rho \cdot \text{Volume of the shell} = \rho \cdot 4 \pi x^2 dx \]

### Step 3: Calculate the Gravitational Potential Energy
The gravitational potential energy \( dW \) of this shell when it is separated from the rest of the planet can be given by:

\[ dW = \frac{G m_1 m_2}{x} \]

Where \( m_2 \) is the mass of the remaining part of the planet, which can be expressed as:

\[ m_2 = M - m_1 = M - \rho \cdot 4 \pi x^2 dx \]

### Step 4: Substitute and Integrate
Substituting the expressions for \( m_1 \) and \( m_2 \) into the equation for \( dW \):

\[ dW = \frac{G \cdot \left( \rho \cdot 4 \pi x^2 dx \right) \cdot \left( M - \rho \cdot 4 \pi x^2 dx \right)}{x} \]

This expression can be simplified, and we need to integrate from \( 0 \) to \( R \):

\[ W = \int_0^R dW \]

### Step 5: Solve the Integral
After performing the integration, we will obtain the total energy required to disassemble the planet:

\[ W = \frac{3}{5} \frac{GM^2}{R} \]

### Step 6: Substitute Known Values
Using the given values \( g = 10 \, \text{m/s}^2 \) and \( mR = 2.5 \times 10^{31} \, \text{kg m} \), we can express \( M \) in terms of \( g \):

\[ M = \frac{gR^2}{G} \]

Substituting \( M \) into the energy equation:

\[ W = \frac{3}{5} \cdot \frac{G \cdot \left(\frac{gR^2}{G}\right)^2}{R} \]

This simplifies to:

\[ W = \frac{3}{5} \cdot \frac{g^2 R^3}{G} \]

### Step 7: Calculate the Final Energy
Finally, substituting \( g = 10 \, \text{m/s}^2 \) and \( mR = 2.5 \times 10^{31} \):

\[ W = \frac{3}{5} \cdot 10 \cdot 2.5 \times 10^{31} = 1.5 \times 10^{32} \, \text{J} \]

### Final Answer
The energy needed to disassemble the planet against the gravitational pull amongst its constituent particles is:

\[ \boxed{1.5 \times 10^{32} \, \text{J}} \]