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The problem asks to find the magnitude of the force \( F \) required to push a block of mass \( m = 15.0 \, \text{kg} \) up an incline at a constant velocity, where the incline has an angle \( \theta = 30^\circ \), and the coefficient of kinetic friction is \( \mu_k = 0.375 \).

### Breakdown of Forces:
1. **Gravitational force** \( F_g = mg \), where \( g \approx 9.8 \, \text{m/s}^2 \).
   - The component of gravity parallel to the incline is \( F_g \sin\theta = mg\sin\theta \).
   - The component of gravity perpendicular to the incline is \( F_g \cos\theta = mg\cos\theta \).

2. **Normal force** \( N \): Since there is no acceleration perpendicular to the incline, the normal force is balanced by the perpendicular component of gravity:
   \[
   N = mg \cos\theta
   \]

3. **Frictional force** \( F_f \): The kinetic friction opposes the motion and is given by \( F_f = \mu_k N \). Substituting \( N \):
   \[
   F_f = \mu_k mg \cos\theta
   \]

4. **Total force required \( F \)**: To maintain constant velocity, the applied force \( F \) must balance both the force due to gravity (parallel to the incline) and the frictional force. The equation for the required force is:
   \[
   F = F_g \sin\theta + F_f
   \]
   Substituting the expressions for \( F_g \) and \( F_f \):
   \[
   F = mg \sin\theta + \mu_k mg \cos\theta
   \]

Now, let’s calculate the magnitude of \( F \).The magnitude of the force \( F \) required to push the block up the incline at a constant velocity is approximately \( 121.24 \, \text{N} \).

Would you like further details on any part of the solution, or have any specific questions?

Here are five related questions to expand your understanding:
1. How would the force change if the incline angle increased?
2. What happens to the frictional force if the surface becomes smoother (lower \( \mu_k \))?
3. How would the required force change if the mass of the block doubled?
4. What would happen to the force if the block was moving down the incline at a constant velocity?
5. How does the constant velocity assumption simplify the problem?

**Tip:** In problems involving inclines, always break forces into components parallel and perpendicular to the incline.

A block of mass m = 15.0 kg moves up an incline with constant velocity while being pushed by a force F. The kinetic coefficient of friction between the block and the floor is μk = 0.375, and the incline makes an angle of θ = 30.0° with respect to the horizontal. What is the magnitude of F?

Solve a physics problem involving a 15 kg block moving up an incline with constant velocity. Learn how to calculate the force required using concepts like Newton's Laws, friction, and inclined planes. Perfect for high school students.

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