The height of the point vertically above the earth's surface, at which acceleration due to gravtiy becomes 1% of its value at the surface is (Radius of the earth =R)

Options

A. 8R
B. 9R
C. 10R
D. 20R

Answer

To find the height \( h \) above the Earth's surface where the acceleration due to gravity becomes 1% of its value at the surface, we can follow these steps:

### Step 1: Write the formula for acceleration due to gravity at the surface
The acceleration due to gravity at the surface of the Earth is given by:
\[ g = \frac{GM}{R^2} \]
where \( G \) is the gravitational constant, \( M \) is the mass of the Earth, and \( R \) is the radius of the Earth.

### Step 2: Write the formula for acceleration due to gravity at height \( h \)
At a height \( h \) above the Earth's surface, the acceleration due to gravity \( g' \) is given by:
\[ g' = \frac{GM}{(R + h)^2} \]

### Step 3: Set up the equation for 1% of surface gravity
We need to find the height \( h \) where \( g' \) is 1% of \( g \):
\[ g' = 0.01g \]
Substituting the expressions for \( g \) and \( g' \):
\[ \frac{GM}{(R + h)^2} = 0.01 \cdot \frac{GM}{R^2} \]

### Step 4: Cancel \( GM \) from both sides
Since \( GM \) is common on both sides, we can cancel it out:
\[ \frac{1}{(R + h)^2} = 0.01 \cdot \frac{1}{R^2} \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ (R + h)^2 = 100R^2 \]

### Step 6: Take the square root
Taking the square root of both sides:
\[ R + h = 10R \]

### Step 7: Solve for \( h \)
Now, solving for \( h \):
\[ h = 10R - R = 9R \]

### Conclusion
Thus, the height \( h \) above the Earth's surface at which the acceleration due to gravity becomes 1% of its value at the surface is:
\[ h = 9R \]