PLEASE ANSWER NUMBER 55 ONLY. I POSTED 54 BECAUSE IT\'S NEEDED TO SOLVE 55. PLEA

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PLEASE ANSWER NUMBER 55 ONLY. I POSTED 54 BECAUSE IT'S NEEDED TO SOLVE 55. PLEASE SHOW THE COMPLETE SOLUTION. THANK YOU. :)

54.       A 2.00-kg block situated on a rough incline is connected to a spring of negligible mass having a spring constant of 100 N/m (Fig. P8.54). The pulley is frictionless. The block is released from rest when the spring is unstretched. The block moves 20.0 cm down the incline before coming to rest. Find the coefficient of kinetic friction between block and incline.

Figure P8.54 Problems 54 and 55.

55.       Review problem. Suppose the incline is frictionless for the system described in Problem 54 (Fig. P8.54). The block is released from rest with the spring initially unstretched. (a) How far does it move down the incline before coming to rest? (b) What is its acceleration at its lowest point? Is the acceleration constant? (c) Describe the energy transformations that occur during the descent.

Explanation / Answer

a)
As no nonconservative work is done so E=0,alsoK=0

Therefore form conservation of energy we have Ui=Uf

Where Ui=(mgsin)x and Uf = 1/2*kx2

So we have (mgsin)x =  1/2*kx2

             or (mgsin) = 1/2*kx

                 (2.00-kg)(9.80 m/s2)sin370 = 1/2*(100N/m)x

                  x = 11.7955/50

                    = 0.2359 m

b)

We have F = ma

as only gravity and spring force act on the block we have

-kx+mgsin =ma

-(100 N/m)(0.2359m)+(2.0kg)(9.80 m/s2)sin370= (2.0 kg)a

-23.59+11.7955 = (2.0)a

     a =-11.7945/2.0

         = -5.89725 m/s2

c)

Ugravity decreases monotonically as the height decreases

Uspring increases monotonically as the spring is streched

K initially increases,but then goes back to zero

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