Find the force required to pull a block of mass 100kg up an incline of `8^(@)` at a uniform speed of `5m//s`. The coefficient of friction between the block and the plane is `0.3`.

Explanation

To find the force required to pull a block of mass 100 kg up an incline of 8 degrees at a uniform speed of 5 m/s, we need to consider the forces acting on the block. Here's the step-by-step solution:

### Step 1: Identify the forces acting on the block
1. **Weight of the block (mg)**: This acts vertically downward. For a mass (m) of 100 kg, the weight is:
\[ W = mg = 100 \, \text{kg} \times 10 \, \text{m/s}^2 = 1000 \, \text{N} \]

2. **Normal force (N)**: This acts perpendicular to the inclined plane.

3. **Frictional force (F_f)**: This opposes the motion and acts down the incline.

4. **Component of weight along the incline (mg sin θ)**: This acts down the incline.

5. **Component of weight perpendicular to the incline (mg cos θ)**: This acts perpendicular to the incline.

### Step 2: Calculate the components of the weight
- The angle of incline (θ) is 8 degrees. We need to calculate:
- \( mg \sin(8^\circ) \)
- \( mg \cos(8^\circ) \)

Using trigonometric values:
- \( \sin(8^\circ) \approx 0.1392 \)
- \( \cos(8^\circ) \approx 0.9903 \)

Calculating the components:
\[ mg \sin(8^\circ) = 1000 \times 0.1392 = 139.2 \, \text{N} \]
\[ mg \cos(8^\circ) = 1000 \times 0.9903 = 990.3 \, \text{N} \]

### Step 3: Calculate the frictional force
The frictional force (F_f) can be calculated using the coefficient of friction (μ) and the normal force (N):
\[ F_f = \mu N = \mu (mg \cos(8^\circ)) \]
Substituting the values:
\[ F_f = 0.3 \times 990.3 \approx 297.09 \, \text{N} \]

### Step 4: Set up the equation of motion
Since the block is moving at a uniform speed, the net force along the incline is zero. Therefore, the pulling force (F) must balance the sum of the frictional force and the component of weight down the incline:
\[ F = mg \sin(8^\circ) + F_f \]
Substituting the values:
\[ F = 139.2 + 297.09 \approx 436.29 \, \text{N} \]

### Final Answer
The force required to pull the block up the incline at a uniform speed is approximately:
\[ F \approx 436.29 \, \text{N} \]