The value of 'g' on the surface of the earth is `9.81 ms^(-2)` . Find its value on the surface of the moon . Given mass of earth `6.4 xx 10^(24)` kg , radius of earth = `6.4 xx 10^(6)` m , mass of the moon = `7.4 xx 10^(22)` kg , radius of moon = `1.76 xx 10^(6)` m .

Answer

To find the value of 'g' on the surface of the Moon, we can use the formula for gravitational acceleration:

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

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

### Step 1: Understand the relationship between 'g' on Earth and 'g' on the Moon

We know that:

\[ g_{\text{Earth}} = \frac{GM_{\text{Earth}}}{R_{\text{Earth}}^2} \]

and for the Moon:

\[ g_{\text{Moon}} = \frac{GM_{\text{Moon}}}{R_{\text{Moon}}^2} \]

### Step 2: Set up the ratio of gravitational accelerations

We can express the gravitational acceleration on the Moon in terms of that on Earth:

\[ g_{\text{Moon}} = g_{\text{Earth}} \times \frac{M_{\text{Moon}}}{M_{\text{Earth}}} \times \left(\frac{R_{\text{Earth}}}{R_{\text{Moon}}}\right)^2 \]

### Step 3: Substitute the known values

Given:
- \( g_{\text{Earth}} = 9.81 \, \text{m/s}^2 \)
- \( M_{\text{Earth}} = 6.4 \times 10^{24} \, \text{kg} \)
- \( R_{\text{Earth}} = 6.4 \times 10^{6} \, \text{m} \)
- \( M_{\text{Moon}} = 7.4 \times 10^{22} \, \text{kg} \)
- \( R_{\text{Moon}} = 1.76 \times 10^{6} \, \text{m} \)

### Step 4: Calculate the ratio of masses and the square of the ratio of radii

1. Calculate the mass ratio:

\[ \frac{M_{\text{Moon}}}{M_{\text{Earth}}} = \frac{7.4 \times 10^{22}}{6.4 \times 10^{24}} = \frac{7.4}{640} = 0.0115625 \]

2. Calculate the radius ratio squared:

\[ \left(\frac{R_{\text{Earth}}}{R_{\text{Moon}}}\right)^2 = \left(\frac{6.4 \times 10^{6}}{1.76 \times 10^{6}}\right)^2 = \left(\frac{6.4}{1.76}\right)^2 = (3.63636)^2 \approx 13.2 \]

### Step 5: Substitute these values into the equation for \( g_{\text{Moon}} \)

Now we can calculate \( g_{\text{Moon}} \):

\[ g_{\text{Moon}} = 9.81 \times 0.0115625 \times 13.2 \]

Calculating this gives:

\[ g_{\text{Moon}} \approx 9.81 \times 0.0115625 \times 13.2 \approx 1.63 \, \text{m/s}^2 \]

### Final Answer

Thus, the value of 'g' on the surface of the Moon is approximately:

\[ g_{\text{Moon}} \approx 1.63 \, \text{m/s}^2 \]

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