15) A o 46-kg block is held in place against the spring by a 30-N horizontal ext

Question

15) A o 46-kg block is held in place against the spring by a 30-N horizontal external force. The external force is removed, and the is projected with velocity 1.2 m/s upon separation from the spring, as shown in the figure. The block descends a ramp and has a velocity v2 1.5 m/s at the bottom. The track is frictionless between points A and B. The block enters a rough section at point B, extending to E. The coefficient of kinetic friction between the block and the rough surface is 0.38. The velocity of the block is v3- 1.4 point The block moves on to D, where it stops. What distance does the block travel between points B and D? What is the height ofthe ramp, h? What is the spring constant (stiffness) of the spring? Smooth B Rough

Explanation / Answer

F = external force = 30 N

k = spring constant

x = compression of spring

V1 = speed of block after leaving spring = 1.2 m/s

m = mass of block = 0.46 kg

external force balances the spring force when block is held compressed against the spring , hence

F = kx

or x = F/k = 30/k

Using conservation of energy

Kinetic energy gained by Block = spring energy

(0.5) m V12 = (0.5) k x2

(0.46) (1.2) = k (30/k)2

0.552 = 900 /k

k = 1630.43 N/m

B to D motion :;

uk = Coefficient of kinetic friction = 0.38

frictional force acting on the block ::

f = uk mg = 0.38 x 0.46 x 9.8 = 1.713 N

work done by frictional force = change in Kinetic energy

f d = (0.5) m V22

1.713 d = (0.5) (0.46) (1.5)2

d = 0.302 m

using conservation of energy :;

(0.5) m V12 + mgh = (0.5) m V22

1.22 + 2 x 9.8 x h = 1.52

h = 0.0413 m

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