A 2.0 kg block is projected by a spring along a frictionless track to the portio

Question

A 2.0 kg block is projected by a spring along a frictionless track to the portion of track with friction at the higher level. The spring constant is k = 200 N/m, and the initial compression is x = 1.2 m. The height difference of the two ends of the track is Ay= 5.0 m and the coefficient of kinetic friction between the block and upper surface is mu k = 0.6. Assume Ug1 = 0 at the initial level. Find: a) the initial spring potential energy, U1, stored in the spring before it is released, b) the initial total mechanical energy, E1, of the block-spring system, c) the kinetic energy, K, of the block when it leaves the spring, d) the gravitational potential energy, Ug2, when the block reaches the new height. e) the kinetic energy of the block, K2, at the new height, just before it begins sliding with friction, f) the amount of work done by friction to stop the block, and, g) the distance,delta x, the block slides along the upper surface before stopping.

Explanation / Answer

a)
Us1 = 0.5*k*x^2
= 0.5*200 * (1.2)^2
= 144 J

b)
Initial total mechanical energy = Us1 = 144 J

c)
Use conservation ofenergy
Kinetic energy of block = spring potential energy
K = Us1
= 144 J

d)
Ug2 = m*g* delta Y
= 2*9.8*5
= 98 J

e)
Use conservation of energy
Kinetic energy at bottom = potential energy at top + kinetic neergy at top
K = Ug2 + K2
144 = 98 + K2
K2= 46 J

f)
Work done by friction = - K2
= -46 J

g)
frictional force = miuk*m*g = 0.6*2*9.8 = 11.76 N
f*delta x = 46
11.76 * delta x = 46
delta x = 3.9 m

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