The upper portion of an inclined plane of inclination a is smooth and the lower portion is rough. A partical slide down from rest the top and just comes to rest at the foot . If the ratio of smooth length to rough length is `m:n`, find the coefficient of friction .

Explanation

To solve the problem step by step, we need to analyze the motion of the particle on the inclined plane, which consists of a smooth upper portion and a rough lower portion.

### Step 1: Define the problem
Let the length of the smooth portion of the inclined plane be \( S_1 = m \) and the length of the rough portion be \( S_2 = n \). The angle of inclination is \( \alpha \).

### Step 2: Apply the equations of motion
Since the particle starts from rest and comes to rest at the foot of the incline, we can use the equations of motion.

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