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A body starts from rest at the top of a rough inclined plane that makes an angle alpha with the horizontal. The coefficient of kinetic friction between the body and the surface varies with distance x measured from the top as mu = b*x, where b is a positive constant. Find the distance from the top at which the body momentarily comes to rest.
- 2 sin(alpha) / b
- 2 cos(alpha) / b
- tan(alpha) / b
- 2 tan(alpha) / b
Correct answer: 2 tan(alpha) / b
Solution
The net force gives a(x) = g*sin(alpha) - g*b*x*cos(alpha). Integrating v dv/dx from 0 to 0 over distance d yields g*sin(alpha)*d - g*b*cos(alpha)*d²/2 = 0, so d = 2*sin(alpha)/(b*cos(alpha)) = 2*tan(alpha)/b.
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