6.) A 0.58-kg object connected to a light spring with a force constant of 18.4 N

Question

6.) A 0.58-kg object connected to a light spring with a force constant of 18.4 N/m oscillates on a frictionless horizontal surface. The spring is compressed 4.0 cm and released from rest.

(a) Determine the maximum speed of the object. ___m/s

(b) Determine the speed of the object when the spring is compressed 1.5 cm.___m/s

(c) Determine the speed of the object as it passes the point 1.5 cm from the equilibrium position.___m/s

(d) For what value of x does the speed equal one-half the maximum speed?___m

Please show step by step

Explanation / Answer

a)

m = mass of object = 0.58 kg

k = spring constant = 18.4 N/m

xmax = compression = 0.04 m

a)

using conservation of energy

maximum kinetic energy = maximum spring potential energy

(0.5) m Vmax2 = (0.5) k xmax2

m Vmax2 = k x2

(0.58) V2 = (18.4) (0.04)2

V = 0.225 m/s

b)

x = compression of spring = 1.5 cm = 0.015 m

Total energy = spring potential energy + kinetic energy

(0.5) k xmax2 = (0.5) k x2 + (0.5) m V2

(18.4) (0.04)2 = (18.4) (0.015)2 + (0.58) V2

V = 0.21 m/s

c)

V = 0.21 m/s

d)

V = 0.225/2 = 0.1125 m/s

Total energy = spring potential energy + kinetic energy

(0.5) k xmax2 = (0.5) k x2 + (0.5) m V2

(18.4) (0.04)2 = (18.4) x2 + (0.58) (0.1125)2

V = 0.035 m/s

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