Given this physics problem, let's go through the calculations step by step to determine the coefficient of kinetic friction between the wooden crate and the inclined plane.

Problem Analysis

We have:

• Mass of the crate, $m = 60 \, \text{kg}$
• Applied force, $F = 800 \, \text{N}$
• Incline height, $h = 2 \, \text{m}$
• Length of the incline, $L = 2.5 \, \text{m}$
• Acceleration of the crate, $a = 2 \, \text{m/s}^2$
• Gravitational acceleration, $g = 10 \, \text{m/s}^2$

We are tasked with finding the coefficient of kinetic friction $\mu_k$.

Step 1: Calculate the Inclination Angle

The incline angle $\theta$ can be determined from the height and the length of the incline using trigonometry: $\sin \theta = \frac{h}{L} = \frac{2}{2.5} = 0.8$ $\theta = \arcsin(0.8)$

Step 2: Resolve Forces along the Incline

The forces acting on the crate along the inclined plane are:

1. The gravitational component along the incline: $F_{\text{gravity}} = mg \sin \theta$
2. The applied force $F = 800 \, \text{N}$
3. The frictional force $F_{\text{friction}} = \mu_k N$, where $N$ is the normal force.

Step 3: Calculate the Normal Force

The normal force $N$ is given by: $N = mg \cos \theta$

Step 4: Apply Newton's Second Law along the Incline

Using Newton's second law, we get: $F - F_{\text{gravity}} - F_{\text{friction}} = ma$ Substituting in the expressions: $800 - (mg \sin \theta) - (\mu_k \cdot mg \cos \theta) = ma$

Step 5: Solve for $\mu_k$

Isolating $\mu_k$, we have: $\mu_k = \frac{F - mg \sin \theta - ma}{mg \cos \theta}$

Let’s substitute the known values and calculate $\mu_k$:

1. $mg = 60 \times 10 = 600 \, \text{N}$
2. Using $\sin \theta = 0.8$ and $\cos \theta = \sqrt{1 - 0.8^2} = 0.6$
3. Plugging these values into the formula for $\mu_k$:

$\mu_k = \frac{800 - 600 \times 0.8 - 60 \times 2}{600 \times 0.6}$

Calculating each term individually:

• $600 \times 0.8 = 480$
• $60 \times 2 = 120$
• $600 \times 0.6 = 360$

So, $\mu_k = \frac{800 - 480 - 120}{360} = \frac{200}{360} = \frac{5}{9} \approx 0.56$

Final Answer

The coefficient of kinetic friction $\mu_k \approx 0.56$.

Would you like further details on any of these steps or have additional questions?

Here are 5 related questions for practice:

1. How would the coefficient of kinetic friction change if the incline length increased?
2. What would be the effect of a larger applied force on the acceleration?
3. How would you calculate the normal force if the incline angle was unknown?
4. What is the role of the frictional force in this problem?
5. How would the solution change if the crate moved down the incline instead?

Tip: When solving physics problems, always break down forces along and perpendicular to the incline to simplify calculations.