 A horizontal force, F1 = 55 N, and a force, F2 = 12.8 N acting at an angle of θ to the horizontal, are applied to a block of mass m = 3.1 kg. The coefficient of kinetic friction between the block and the surface is μk = 0.2. The block is moving to the right.

Randomized Variables
F1 = 55 N
F2 = 12.8 N
m = 3.1 kg


      Part (a)
Solve numerically for the magnitude of the normal force, FN in Newtons, that acts on the block if θ = 30°. 
FN 

To solve for the magnitude of the normal force \( F_N \) acting on the block, we will analyze the forces acting on it. The normal force \( F_N \) is the force exerted by the surface to balance the vertical forces acting on the block.

### Forces acting on the block:
1. The **weight** of the block, \( W = mg \), acts downward, where:
   \[
   W = m \cdot g = 3.1 \, \text{kg} \cdot 9.8 \, \text{m/s}^2
   \]
   This is the gravitational force.
   
2. The **force \( F_2 \)** has both vertical and horizontal components due to the angle \( \theta = 30^\circ \):
   - The vertical component of \( F_2 \) is \( F_2 \sin \theta \), which acts upward and reduces the normal force.
   
3. The **normal force \( F_N \)**, which acts upward, opposes the net vertical force to prevent the block from moving in the vertical direction.

### Calculation of \( F_N \):

The normal force balances the net vertical forces. This gives us the following equation:
\[
F_N = W - F_2 \sin \theta
\]
Where:
- \( W = mg \) is the weight of the block,
- \( F_2 \sin \theta \) is the upward vertical component of force \( F_2 \).

Now, let’s calculate this step-by-step:

1. Compute the weight of the block:
\[
W = 3.1 \, \text{kg} \cdot 9.8 \, \text{m/s}^2 = 30.38 \, \text{N}
\]

2. Calculate the vertical component of \( F_2 \):
\[
F_2 \sin 30^\circ = 12.8 \, \text{N} \cdot \sin 30^\circ = 12.8 \, \text{N} \cdot 0.5 = 6.4 \, \text{N}
\]

3. Substitute into the normal force equation:
\[
F_N = 30.38 \, \text{N} - 6.4 \, \text{N} = 23.98 \, \text{N}
\]

Thus, the magnitude of the normal force \( F_N \) is approximately:
\[
F_N \approx 23.98 \, \text{N}
\]

Would you like more details on any step, or have any questions?

Here are 5 related questions for further exploration:
1. What is the frictional force acting on the block?
2. How would the normal force change if \( \theta \) were increased?
3. What is the net horizontal force acting on the block?
4. How does the coefficient of kinetic friction affect the motion of the block?
5. How would the normal force be affected if the mass of the block increased?

**Tip:** The vertical component of a force at an angle can significantly reduce the normal force, thus affecting the friction.

A horizontal force, F1 = 55 N, and a force, F2 = 12.8 N acting at an angle of θ to the horizontal, are applied to a block of mass m = 3.1 kg. The coefficient of kinetic friction between the block and the surface is μk = 0.2. The block is moving to the right. Solve numerically for the magnitude of the normal force, FN in Newtons, that acts on the block if θ = 30°.

Learn how to calculate the normal force acting on a block with mass 3.1 kg under the influence of two forces: F1 = 55 N horizontally and F2 = 12.8 N at a 30° angle to the horizontal. This solution involves concepts like force components, weight, and friction, suitable for high school students in Grades 10-12.

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