A pushing force of F, at angle theta is applied to a box of mass m. If the box moves at a constant velocity, what is the coefficient of Friction in terms of f, m, and theta

To find the coefficient of friction in terms of \( F \), \( m \), and \( \theta \) when a box is pushed at a constant velocity, we can start by analyzing the forces acting on the box.

### Forces Acting on the Box:

1. **Pushing Force \( F \)**:
   - Horizontal component: \( F_x = F \cos \theta \)
   - Vertical component: \( F_y = F \sin \theta \)

2. **Weight \( mg \)**:
   - Acts vertically downward.

3. **Normal Force \( N \)**:
   - Acts vertically upward, opposing the weight and the vertical component of the pushing force.
   - \( N = mg - F \sin \theta \)

4. **Frictional Force \( f_{\text{friction}} \)**:
   - Opposes the horizontal component of the pushing force.
   - \( f_{\text{friction}} = \mu N \), where \( \mu \) is the coefficient of friction.

Since the box moves at a constant velocity, the net force in the horizontal direction is zero, meaning:
\[
F \cos \theta = f_{\text{friction}}
\]

Substitute the expression for the frictional force:
\[
F \cos \theta = \mu N
\]

We already know that:
\[
N = mg - F \sin \theta
\]

Substitute this into the equation:
\[
F \cos \theta = \mu (mg - F \sin \theta)
\]

Now, solving for \( \mu \):
\[
\mu = \frac{F \cos \theta}{mg - F \sin \theta}
\]

### Coefficient of Friction in Terms of \( f \), \( m \), and \( \theta \)

The problem asks for the coefficient of friction in terms of the frictional force \( f_{\text{friction}} \), mass \( m \), and angle \( \theta \).

Since \( F \cos \theta = f_{\text{friction}} \), we can replace \( F \cos \theta \) with \( f \) in the previous equation:
\[
\mu = \frac{f}{mg - F \sin \theta}
\]

Thus, the coefficient of friction \( \mu \) in terms of \( f \), \( m \), and \( \theta \) is:
\[
\boxed{\mu = \frac{f}{mg - F \sin \theta}}
\]

Would you like more details or have any further questions?

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Here are five related questions to consider:

1. How does the frictional force \( f \) change if the angle \( \theta \) is increased?
2. What happens to the normal force \( N \) when the angle \( \theta \) changes?
3. How would the coefficient of friction change if the pushing force is applied horizontally (i.e., \( \theta = 0^\circ \))?
4. Can this equation be used if the box is accelerating? Why or why not?
5. How does the weight of the box \( mg \) influence the coefficient of friction?

**Tip:** When working with forces at an angle, always resolve them into horizontal and vertical components to simplify the analysis.

Learn how to calculate the coefficient of friction in terms of the applied force f, mass m, and angle theta. Understand the forces involved, including frictional force and normal force, using trigonometric principles. Suitable for advanced high school physics students.

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