A body weighing 20 kg just slides down a rough inclined plane that rises 5 m in every 13 m. What is the coefficient of friction?

Answer

To find the coefficient of friction (μ) for a body weighing 20 kg that just slides down a rough inclined plane, we can follow these steps:

### Step 1: Understand the Geometry of the Incline
The inclined plane rises 5 m for every 13 m of length. We can use this information to find the angle of inclination (θ).

### Step 2: Calculate the Angle of Inclination
We can use the tangent function to find the angle:
\[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{5}{12} \]
Here, the rise is 5 m and the horizontal distance (base) can be found using the Pythagorean theorem:
\[ \text{base} = \sqrt{(13^2 - 5^2)} = \sqrt{169 - 25} = \sqrt{144} = 12 \text{ m} \]

### Step 3: Establish the Forces Acting on the Body
When the body is sliding down, the forces acting on it include:
- Weight (W = mg) acting downward
- Normal force (N) acting perpendicular to the incline
- Frictional force (F_f) acting opposite to the direction of motion

### Step 4: Resolve the Weight into Components
The weight of the body can be resolved into two components:
- Perpendicular to the incline: \( W_{\perpendicular} = mg \cos(\theta) \)
- Parallel to the incline: \( W_{\parallel} = mg \sin(\theta) \)

### Step 5: Apply Newton's Second Law
Since the body is just sliding down, the net force acting along the incline is zero:
\[ W_{\parallel} - F_f = 0 \]
Where \( F_f = \mu N \) and \( N = W_{\perpendicular} \).

### Step 6: Set Up the Equation
From the above, we can write:
\[ mg \sin(\theta) = \mu mg \cos(\theta) \]
Dividing both sides by \( mg \) (since m is not zero):
\[ \sin(\theta) = \mu \cos(\theta) \]
Thus, we can express the coefficient of friction as:
\[ \mu = \frac{\sin(\theta)}{\cos(\theta)} = \tan(\theta) \]

### Step 7: Calculate the Coefficient of Friction
From our earlier calculation, we have:
\[ \tan(\theta) = \frac{5}{12} \]
Thus, the coefficient of friction (μ) is:
\[ \mu = \frac{5}{12} \approx 0.4167 \]

### Final Answer
The coefficient of friction is approximately **0.4167**.

---