The moon's radius is `1//4` that of the earth and its mass `1//80` times that of the earth. If g represents the acceleration due to gravity on the surface of the earth, that on the surface of the moon is

Options

A. `g/4`
B. `g/4`
C. `g/6`
D. `g/8`

Answer

To find the acceleration due to gravity on the surface of the moon, we can use the formula for gravitational acceleration:

\[ g = \frac{GM}{R^2} \]

where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the celestial body,
- \( R \) is the radius of the celestial body.

### Step 1: Write the formula for the acceleration due to gravity on the moon
The acceleration due to gravity on the moon (\( g_{moon} \)) can be expressed as:

\[ g_{moon} = \frac{G \cdot M_{moon}}{R_{moon}^2} \]

### Step 2: Write the formula for the acceleration due to gravity on the earth
The acceleration due to gravity on the earth (\( g_{earth} \)) is given by:

\[ g_{earth} = \frac{G \cdot M_{earth}}{R_{earth}^2} \]

### Step 3: Take the ratio of \( g_{moon} \) to \( g_{earth} \)
Now, we can find the ratio of the gravitational accelerations:

\[ \frac{g_{moon}}{g_{earth}} = \frac{G \cdot M_{moon} / R_{moon}^2}{G \cdot M_{earth} / R_{earth}^2} \]

The \( G \) cancels out:

\[ \frac{g_{moon}}{g_{earth}} = \frac{M_{moon}}{M_{earth}} \cdot \frac{R_{earth}^2}{R_{moon}^2} \]

### Step 4: Substitute the values for mass and radius
Given:
- The mass of the moon is \( \frac{1}{80} \) times the mass of the earth: \( M_{moon} = \frac{1}{80} M_{earth} \)
- The radius of the moon is \( \frac{1}{4} \) times the radius of the earth: \( R_{moon} = \frac{1}{4} R_{earth} \)

Substituting these values into the equation gives:

\[ \frac{g_{moon}}{g_{earth}} = \left(\frac{1}{80}\right) \cdot \left(\frac{R_{earth}^2}{\left(\frac{1}{4} R_{earth}\right)^2}\right) \]

### Step 5: Simplify the equation
Calculating the radius term:

\[ \left(\frac{1}{4} R_{earth}\right)^2 = \frac{1}{16} R_{earth}^2 \]

So, we have:

\[ \frac{g_{moon}}{g_{earth}} = \frac{1}{80} \cdot \frac{R_{earth}^2}{\frac{1}{16} R_{earth}^2} \]

This simplifies to:

\[ \frac{g_{moon}}{g_{earth}} = \frac{1}{80} \cdot 16 \]

### Step 6: Calculate the final value
Now, we can calculate:

\[ \frac{g_{moon}}{g_{earth}} = \frac{16}{80} = \frac{1}{5} \]

### Step 7: Conclusion
Thus, the acceleration due to gravity on the surface of the moon is:

\[ g_{moon} = \frac{g_{earth}}{5} \]

### Final Answer
The acceleration due to gravity on the surface of the moon is \( \frac{g}{5} \).

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