A mass M = 3kg starts from rest and slides down (h = 10m) a frictionless surface

Question

A mass M = 3kg starts from rest and slides down (h = 10m) a frictionless surface and smoothly transitions to a flat surface with coefficient of static friction mu k = 0.4. The mass travels a distance d = 10m on this surface and then hits a spring with k = 400N/m. The surface under the spring is frictionless. What is the speed of the mass at points B? VB = ____ What is die speed of die mass at point C. and how much thermal energy has been generated between B and C? vC = Ethermal = What is the maximum compression of the spring? Delta x max = The spring pushes the mass back towards its starling position Does the spring make it back to the slide (past point B)? If so, how high does the mass make it before mining around? If not. where does the mass stop? SHOW WORK!

Explanation / Answer

(a) The mass, M = 3 kg

The initial velocity of mass, u = 0

Height of the block, h = 10 m

From law of conservation of energy, we have

(1/2)mv^2 = m g h

v = [ 2 g h ] ^ (1/2)

   = 14 m/s

So the velocity at point B is, vb = 14 m/s

(b) The acceleration, a = - g = -0.4*9.8 = - 3.92 m/s^2

The distance, s = 10 m

The velocity at B, vb = 14 m/s

We have, vc^2 - vb^2 = 2 a s

               vc^2 = vb^2 + 2 a s

So the velocity at C, vc = 10.8 m/s

The thermal energy generated = The change in the kinetic energy

                                              = (1/2)mv^2 = 0.5 * 3 * 10.8^2 = 174.96 J

(c) We have

(1/2)kx^2 = (1/2)mvc^2

x^2 = mvc^2 / k = 0.8748

x = 0.94 m

(d) F = k x = 400 * 0.94 = 376

a = F / m = 376 / 3 = 125.3 m/s^2

v^2 - vc^2 = 2 a s

Finally mass should be at rest, so v= 0

s = - vc^2 / 2a = 0.46 m

So it does not go to slide

               

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