**Introduction:**

In this scenario, we have a block of mass 0.1 kg that is hanging over a frictionless light fixed pulley. One end of the string is attached to the block and the other end is being pulled by a constant force F in the vertically downward direction. We are given that the linear momentum of the block increases by 2 kg m/s in 1 second, and the acceleration due to gravity is 10 m/s^2.

**Explanation:**

To understand why the linear momentum of the block increases, we need to analyze the forces acting on it and apply Newton's second law of motion.

**1. Forces acting on the block:**

- Weight (mg): The block experiences a downward force due to its weight, which is equal to the mass (m) multiplied by the acceleration due to gravity (g). In this case, the weight of the block is 0.1 kg * 10 m/s^2 = 1 N.
- Tension (T): The tension in the string is the force transmitted through the string from one end to the other. In this case, the tension in the string is equal to the force (F) pulling the other end of the string.

**2. Newton's second law of motion:**

According to Newton's second law of motion, the net force acting on an object is equal to the mass of the object multiplied by its acceleration. Mathematically, this can be represented as follows:

Net force (F_net) = mass (m) * acceleration (a)

**3. Calculating the acceleration of the block:**

Since the block is connected to a fixed pulley, it experiences the same acceleration as the other end of the string. Therefore, the acceleration of the block can be determined using the tension in the string.

From the forces acting on the block, we can write the following equation:

T - mg = ma

Substituting the values, we get:

F - 1 N = 0.1 kg * a

**4. Solving for the acceleration:**

We are given that the linear momentum of the block increases by 2 kg m/s in 1 second. Linear momentum (p) is the product of mass and velocity. Mathematically, this can be represented as:

p = m * v

Since the block is hanging vertically, its initial velocity is zero. Therefore, the change in momentum is equal to the mass multiplied by the final velocity:

Δp = m * Δv

Δp = 0.1 kg * 2 m/s = 0.2 kg m/s

We know that force (F) is the rate of change of momentum. So, we can write:

F = Δp / Δt

F = 0.2 kg m/s / 1 s = 0.2 N

Substituting this value into the previous equation, we get:

0.2 N - 1 N = 0.1 kg * a

-0.8 N = 0.1 kg * a

a = -0.8 N / 0.1 kg = -8 m/s^2

The negative sign indicates that the acceleration is in the opposite direction to the force applied.

**Conclusion:**

In conclusion, the linear momentum of the block increases by 2 kg m/s in 1 second because a constant force of 0.