A spring of negligible mass stretches 3.00 cm from its relaxed length when a for

Question

A spring of negligible mass stretches 3.00 cm from its relaxed length when a force of 8.50 N is applied. A 0.530-kg particle rests on a frictionless horizontal surface and is attached to the free end of the spring. The particle is displaced from the origin to x = 5.00 cm and released from rest at t = 0. (Assume that the direction of the initial displacement is positive.)

(a) What is the force constant of the spring?
280 N/m

(b) What are the angular frequency (?), the frequency, and the period of the motion?

(c) What is the total energy of the system?
0.35 J

(d) What is the amplitude of the motion?
5 cm

(e) What are the maximum velocity and the maximum acceleration of the particle?

(f) Determine the displacement x of the particle from the equilibrium position at t = 0.500 s.
  
________________ cm

(g) Determine the velocity and acceleration of the particle when t = 0.500 s. (Indicate the direction with the sign of your answer.)

Explanation / Answer

a)k=F/x=283.3N/m

b)w=sqrt(k/m)=23.121rad/s

f=w/2pi=3.68Hz

T=1/f=0.2717s

c)E=0.5kA^2=0.354J

d)5cm

e)vmax=Aw=1.1561m/s

amax=Aw^2=26.73m/s^2

f)x=ACos(wt)=5Cos(23.121*0.5)=2.677cm

g)v=-AwSin(wt)=0.9765m/s

a=-Aw^2Cos(wt)=14.31m/s^2

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