A 2-kg mass is released on a smooth incline 4 meters from a spring(k = 100n/m).

Question

A 2-kg mass is released on a smooth incline 4 meters from a spring(k = 100n/m). The spring is fixed along the plane inclined attheta = 30 degrees.   Part a) Find the maximum compression of thespring, assumed to be massless.   Part b) If the incline is not smooth but thecoefficient of friction between it and the mass is .2, find themaximum compression.   Part c) For the rough incline, how far upthe incline will the mass travel after leaving the spring?

  Part a) Find the maximum compression of thespring, assumed to be massless.   Part b) If the incline is not smooth but thecoefficient of friction between it and the mass is .2, find themaximum compression.   Part c) For the rough incline, how far upthe incline will the mass travel after leaving the spring?

Explanation / Answer

Part a)the forces acting on the mass are: in x direction: Fx = w1 = w *sin ----------(1) in y direction: Fy = n + (-w2) = 0 or n = w2 = w * cos -----------(2) let the maximum compression of the spring be x therefore weget w1 = k * x or w * sin = k * x or x = (w * sin/k) = (mg * sin/k) m = 2-kg,g = 9.8 m/s2, = 30o and k= 100 N/m Part b)when the coefficient of friction between the inclineand the mass is k = .2 then the force acting onthe spring is Fx = w1 - fk =w1 - k * n ------------(3) from equation (2) we get or Fx = w * sin - k *w * cos = w * (sin - k * cos)------------(4) or k * x1 = w * (sin - k *cos) or x1 = (w/k) * (sin - k *cos) Part c)we know that v2 - u2 = 2a * x1-------------(5) v = 0 and the acceleration of the mass is obtained fromequation (4) therefore we get m * a = w * (sin - k * cos) or m * a = m * g * (sin - k *cos) or a = g * (sin - k * cos) substituting the value of a in equation (5) we get - u2 = 2 * g * (sin - k *cos) * x1 or u = -(2 * g * (sin - k *cos) * x1)1/2 the negative value of velocity indicates that the mass movesup the inclined plane. the negative value of velocity indicates that the mass movesup the inclined plane.

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