At what distance from the center of the earth will a 1 kg object have a weight of `1 N` ? If released from rest at this distance, What will its initial acceleration be ?

Answer

To solve the problem, we need to determine the distance from the center of the Earth where a 1 kg object will have a weight of 1 N, and then find its initial acceleration if released from rest at that distance.

### Step-by-Step Solution:

1. **Understanding Weight and Gravitational Acceleration**:
The weight \( W \) of an object is given by the formula:
\[ W = m \cdot g \]
where \( m \) is the mass of the object and \( g \) is the gravitational acceleration. For our case, we have:
\[ W = 1 \, \text{N}, \quad m = 1 \, \text{kg} \]
Therefore, we can write:
\[ 1 \, \text{N} = 1 \, \text{kg} \cdot g \]
From this, we find:
\[ g = 1 \, \text{m/s}^2 \]

2. **Using the Formula for Gravitational Acceleration**:
The gravitational acceleration at a distance \( r \) from the center of the Earth can be expressed as:
\[ g = \frac{G \cdot M}{r^2} \]
where \( G \) is the universal gravitational constant, and \( M \) is the mass of the Earth.

3. **Relating Gravitational Acceleration at the Surface of the Earth**:
The gravitational acceleration at the surface of the Earth \( g_0 \) is given by:
\[ g_0 = \frac{G \cdot M}{R^2} \]
where \( R \) is the radius of the Earth.

4. **Setting Up the Equation**:
We know that at the distance \( r \) where the weight is 1 N:
\[ \frac{G \cdot M}{r^2} = 1 \, \text{m/s}^2 \]
We can relate this to the surface gravity:
\[ \frac{G \cdot M}{r^2} = \frac{G \cdot M}{R^2} \cdot \frac{R^2}{r^2} \]
Simplifying gives:
\[ \frac{g_0}{r^2} = 1 \]
Thus:
\[ r^2 = 10 \cdot R^2 \]

5. **Calculating the Distance \( r \)**:
Taking the square root of both sides, we find:
\[ r = \sqrt{10} \cdot R \]
Given that the radius of the Earth \( R \approx 6400 \times 10^3 \, \text{m} \):
\[ r = \sqrt{10} \cdot (6400 \times 10^3) \approx 2.203 \times 10^7 \, \text{m} \]

6. **Finding Initial Acceleration**:
The initial acceleration \( a \) when the object is released from rest can be calculated using the gravitational acceleration at that distance:
\[ a = g = 1 \, \text{m/s}^2 \]

### Final Answers:
- The distance from the center of the Earth where the weight of the object is 1 N is approximately \( 2.203 \times 10^7 \, \text{m} \).
- The initial acceleration of the object when released from rest at this distance is \( 1 \, \text{m/s}^2 \).