Question

A block of mass m = 3.39 kg is attached to a spring (k = 28.7 N/m) by a rope that hangs over a pulley of mass M = 6.78 kg and radius R = 7.81 cm, as shown in the figure.

a) Treating the pulley as a solid homogeneous disk, neglecting friction at the axle of the pulley, and assuming the system starts from rest with the spring at its natural length, find the speed of the block after it falls 1.00 m. Tries 0/99

b) And find the maximum extension of the spring. Tries 0/99

Answer 1

GIven

mass of block m = 3.39 kg

spring ocnstant of the spring   k = 28.7 N / m

mass of pulley M = 6.78 kg

radius of pulley r = 7.81 cm = 0.0781 m

assuming the system starts from rest with the spring at its natural length,  

According to law of conservation of energy

               0 = (1/2)kx² -mgx +(1/2)I?² + (1/2) m v ²

   moment of inertia of solid disk is I = Mr² / 2  

                0 =   kx² - 2mgx + (Mr² /2) ( v² / r ² ) + mv²

               2 mgx - k x ² = v²   (M /2 + m )

                2(3.39)(9.8)(1) - (28.7) = v ² ( 6.78 /2 + 3.39 )

                              v = 2.359 m/s

b)maximum extension of the spring is at

                 (1/2) kx ²   = m g x  

                            x = 2m g / k

                               = 2 ( 3.39 ) ( 9.8 ) / 28.7

                                = 2.315 m