Assuming earth to be a sphere of uniform mass density, how much would a body weigh half way down the center of the center of the earth , if it weighed 100 N on the surface?

Explanation

To solve the problem of determining how much a body would weigh halfway to the center of the Earth, given that it weighs 100 N on the surface, we can follow these steps:

### Step 1: Understand the relationship between weight and gravity
The weight of an object is given by the formula:
\[ W = mg \]
where \( W \) is the weight, \( m \) is the mass of the object, and \( g \) is the acceleration due to gravity.

### Step 2: Identify the weight on the surface
We know that the weight of the body on the surface of the Earth is:
\[ W = 100 \, \text{N} \]

### Step 3: Calculate the mass of the body
From the weight formula, we can express the mass of the body as:
\[ m = \frac{W}{g} \]
where \( g \) is the acceleration due to gravity at the surface of the Earth (approximately \( 9.8 \, \text{m/s}^2 \)). However, we will not need to calculate \( m \) explicitly for this problem.

### Step 4: Determine the depth
The problem states that the body is halfway to the center of the Earth. Let \( R_e \) be the radius of the Earth. The depth \( D \) at which the body is located is:
\[ D = \frac{1}{2} R_e \]

### Step 5: Calculate the acceleration due to gravity at depth
The formula for the acceleration due to gravity at a depth \( D \) is given by:
\[ g' = g \left(1 - \frac{D}{R_e}\right) \]
Substituting \( D = \frac{1}{2} R_e \):
\[ g' = g \left(1 - \frac{\frac{1}{2} R_e}{R_e}\right) = g \left(1 - \frac{1}{2}\right) = g \left(\frac{1}{2}\right) \]
Thus, the acceleration due to gravity at this depth is:
\[ g' = \frac{g}{2} \]

### Step 6: Calculate the weight at the depth
The weight of the body at the depth \( D \) can now be calculated using the new acceleration due to gravity:
\[ W' = mg' = m \left(\frac{g}{2}\right) = \frac{mg}{2} \]
Since \( mg = 100 \, \text{N} \):
\[ W' = \frac{100 \, \text{N}}{2} = 50 \, \text{N} \]

### Conclusion
The weight of the body halfway to the center of the Earth is:
\[ W' = 50 \, \text{N} \]

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