The radius of the moon is `1//4` th the radius of the earth and its mass is `1//80` th the mass of the earth. Calculate the value of g on the surface of the moon.

Explanation

To calculate the value of \( g \) on the surface of the moon, we can use the formula for acceleration due to gravity:

\[ g = \frac{G \cdot M}{R^2} \]

Where:
- \( G \) is the universal gravitational constant,
- \( M \) is the mass of the celestial body (in this case, the moon),
- \( R \) is the radius of the celestial body (in this case, the moon).

### Step 1: Define the known values
Given:
- The radius of the moon \( R_m = \frac{1}{4} R_e \) (where \( R_e \) is the radius of the Earth).
- The mass of the moon \( M_m = \frac{1}{80} M_e \) (where \( M_e \) is the mass of the Earth).

### Step 2: Substitute the values into the formula
We can express the acceleration due to gravity on the moon \( g_m \) as:

\[ g_m = \frac{G \cdot M_m}{R_m^2} \]

Substituting the values of \( M_m \) and \( R_m \):

\[ g_m = \frac{G \cdot \left(\frac{1}{80} M_e\right)}{\left(\frac{1}{4} R_e\right)^2} \]

### Step 3: Simplify the expression
The denominator becomes:

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

Thus, we can rewrite \( g_m \):

\[ g_m = \frac{G \cdot \left(\frac{1}{80} M_e\right)}{\frac{1}{16} R_e^2} \]

This simplifies to:

\[ g_m = \frac{G \cdot M_e}{R_e^2} \cdot \frac{1}{80} \cdot 16 \]

### Step 4: Factor out \( g_e \)
We know that:

\[ g_e = \frac{G \cdot M_e}{R_e^2} \]

Substituting this into our equation gives:

\[ g_m = g_e \cdot \frac{16}{80} \]

### Step 5: Simplify the fraction
Now, simplify \( \frac{16}{80} \):

\[ \frac{16}{80} = \frac{1}{5} \]

Thus, we have:

\[ g_m = \frac{1}{5} g_e \]

### Step 6: Substitute the value of \( g_e \)
Given that \( g_e \approx 9.8 \, \text{m/s}^2 \):

\[ g_m = \frac{1}{5} \cdot 9.8 \approx 1.96 \, \text{m/s}^2 \]

### Final Answer
Therefore, the value of \( g \) on the surface of the moon is approximately:

\[ g_m \approx 1.96 \, \text{m/s}^2 \]