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

If a box is moving at a constant velocity, the net force acting on it must be zero. This means that the pushing force and frictional force must balance each other out.

Let's break down the forces acting on the box:

1. **Pushing Force**: A force \( F \) is applied at an angle \( \theta \) to the horizontal. The force can be resolved into two components:
   - Horizontal component: \( F_x = F \cos \theta \)
   - Vertical component: \( F_y = F \sin \theta \)

2. **Normal Force**: The normal force \( N \) is the force exerted by the surface perpendicular to the box. It is affected by both the weight of the box and the vertical component of the pushing force. The normal force is given by:
   \[
   N = mg - F \sin \theta
   \]
   where \( mg \) is the gravitational force (weight) acting downward.

3. **Frictional Force**: The frictional force \( f_{\text{friction}} \) opposes the motion and is given by:
   \[
   f_{\text{friction}} = \mu N
   \]
   where \( \mu \) is the coefficient of friction.

Since the box moves at a constant velocity, the horizontal forces must balance out. Thus:
\[
F \cos \theta = f_{\text{friction}} = \mu N
\]

Substituting the expression for the normal force \( N \) into the equation:
\[
F \cos \theta = \mu (mg - F \sin \theta)
\]

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

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

Would you like any details on this solution or further explanation?

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Here are five related questions to consider:

1. How does increasing the angle \( \theta \) affect the coefficient of friction?
2. What happens to the coefficient of friction if the pushing force \( F \) is increased?
3. How would the scenario change if the surface was inclined instead of horizontal?
4. How can this equation be applied to determine the required force to move an object at a constant velocity?
5. What role does the mass \( m \) play in the coefficient of friction when the applied force is at an angle?

**Tip:** Always resolve forces into their components when dealing with forces applied at an angle. This helps in understanding how each force contributes to the overall motion.

Explore how to calculate the coefficient of friction when a box of mass m is pushed with force F at angle theta, moving at constant velocity. Understand the physics behind balancing horizontal forces, normal force calculation, and frictional force. Suitable for high school students studying force and motion.

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