To calculate the gravitational force applied by the Moon on an object of 10 kg present on the surface of the Earth, we can use Newton's law of universal gravitation.

Gravitational Force Formula
The gravitational force \( F \) between two masses can be calculated using the formula:
\[ F = \frac{G \cdot m_1 \cdot m_2}{r^2} \]
Where:
- \( F \) = gravitational force
- \( G \) = gravitational constant \( (6.674 \times 10^{-11} \, \text{N m}^2/\text{kg}^2) \)
- \( m_1 \) = mass of the Moon \( (7.35 \times 10^{22} \, \text{kg}) \)
- \( m_2 \) = mass of the object \( (10 \, \text{kg}) \)
- \( r \) = distance from the center of the Moon to the object (approximately \( 3.84 \times 10^8 \, \text{m} \))

Calculation Steps
1. **Substitute the values into the formula:**
\[ F = \frac{(6.674 \times 10^{-11}) \cdot (7.35 \times 10^{22}) \cdot (10)}{(3.84 \times 10^8)^2} \]
2. **Calculate the denominator:**
\[ (3.84 \times 10^8)^2 = 1.47456 \times 10^{17} \]
3. **Now calculate \( F \):**
\[ F = \frac{(6.674 \times 10^{-11}) \cdot (7.35 \times 10^{22}) \cdot (10)}{1.47456 \times 10^{17}} \]
\[ F \approx \frac{4.9034 \times 10^{12}}{1.47456 \times 10^{17}} \approx 3.32 \times 10^{-5} \, \text{N} \]

Conclusion
The gravitational force applied by the Moon on a 10 kg object on the surface of the Earth is approximately \( 3.32 \times 10^{-5} \, \text{N} \). This force is very small compared to the gravitational force exerted by the Earth on the same object, which is approximately \( 98 \, \text{N} \).