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Because the block is not moving, the sum of the x components of the forces acting on the block must be zero. Find an expression for the sum of the x components of the forces acting on the block, using coordinate system b.
Express your answer in terms of some or all of the variables  Fn
 ,  Ff
 ,  Fw
 , and  θ
 .
View Available Hint(s)for Part D
Hint 1for Part D. Find the x component of  F⃗ n
 
Write an expression for  Fnx
 , the x component of the force  F⃗ n
 , using coordinate system b.
Express your answer in terms of  Fn
  and  θ
 .
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Fnx
  =



]
∑Fx=0
  =

In the image, a block rests on an inclined plane with an angle \(\theta\). The forces acting on the block are:

- \( \vec{F}_n \): Normal force perpendicular to the plane.
- \( \vec{F}_f \): Frictional force parallel to the plane, opposing motion.
- \( \vec{F}_w \): Gravitational force acting vertically downward.

We are tasked with finding the expression for the sum of the forces in the x-direction using coordinate system "b", which has its x-axis aligned with the inclined plane (parallel to it).

### Components of Forces in the x-direction (parallel to the incline):

1. **Normal Force (\(\vec{F}_n\)):**
   Since the normal force is perpendicular to the incline, its x-component is:
   \[
   F_{nx} = -F_n \sin(\theta)
   \]
   The negative sign indicates that the x-component of \( \vec{F}_n \) is directed in the opposite direction along the x-axis.

2. **Gravitational Force (\(\vec{F}_w\)):**
   The gravitational force can be decomposed into two components: one parallel to the incline (x-direction) and one perpendicular (y-direction). The x-component of the weight (gravitational force) is:
   \[
   F_{wx} = -F_w \sin(\theta)
   \]
   This is the component of the weight that acts down the slope.

3. **Frictional Force (\(\vec{F}_f\)):**
   The frictional force acts along the incline, opposing motion, so:
   \[
   F_{fx} = F_f
   \]
   No need to decompose it as it acts purely along the x-axis.

### Sum of Forces in the x-direction:

Since the block is not moving, the sum of forces in the x-direction must be zero:
\[
\sum F_x = F_{nx} + F_{fx} + F_{wx} = 0
\]
Substituting the components:
\[
- F_n \sin(\theta) + F_f - F_w \sin(\theta) = 0
\]
Simplifying:
\[
F_f = F_n \sin(\theta) + F_w \sin(\theta)
\]

This is the expression for the sum of the x-components of the forces acting on the block.

### Do you have any further questions on this solution? Here are some related questions to deepen your understanding:

1. How would you calculate the normal force \( F_n \) in terms of \( F_w \) and \( \theta \)?
2. If friction were absent, what would the equation of motion be along the incline?
3. How does increasing the angle \( \theta \) affect the frictional force required to keep the block stationary?
4. What happens to the normal force if the block were to start sliding?
5. How could you express the frictional force \( F_f \) in terms of the coefficient of friction and the normal force?

**Tip:** To ensure the block stays stationary, remember that the frictional force must equal the component of the gravitational force pulling the block down the incline.

Because the block is not moving, the sum of the x components of the forces acting on the block must be zero. Find an expression for the sum of the x components of the forces acting on the block, using coordinate system b. Express your answer in terms of some or all of the variables Fn, Ff, Fw, and θ.

Explore how to find the sum of the x-components of forces acting on a block resting on an inclined plane. Learn to decompose forces like the normal force and gravitational force, applying Newton's First Law for static equilibrium. Suitable for high school students studying physics.

Math Problem Statement

Fnx

] ∑Fx=0

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