A 2.50-kg object is attached to a spring and placed on frictionless, horizontal

Question

A 2.50-kg object is attached to a spring and placed on frictionless, horizontal surface. A horizontal force of 16.0 N is required to hold the object at rest when it is pulled 0.200 m from its equilibrium position (the origin of the x axis). The object is now released from rest from this stretched position, and it subsequently undergoes simple harmonic oscillations.

(a) Find the force constant of the spring.
N/m

(b) Find the frequency of the oscillations.
Hz

(c) Find the maximum speed of the object.
m/s

(d) Where does this maximum speed occur?
x = ±  m

(e) Find the maximum acceleration of the object.
m/s2

(f) Where does the maximum acceleration occur?
x = ±  m

(g) Find the total energy of the oscillating system.
J

(h) Find the speed of the object when its position is equal to one-third of the maximum value.
m/s

(i) Find the magnitude of the acceleration of the object when its position is equal to one-third of the maximum value.
m/s2

Explanation / Answer

a) F = Kx
   16 = K *0.2

force constant , K = 80 N/m
b)

angular frequency ,w = sqrt(k/m) = sqrt(80/2.50) = 5.656 rad/s

frequency = w/2pi = 0.9 Hz

c) maximum speed = w*A = 5.656*0.2 = 1.1312 m/s

d) maximum speed occurs at +/- 0 m

e) maximum acceleration ,a = A*w^2 = 6.398 m/s^2

f) the maximum acceleration occur at +/- 0.2 m

g) total energy = 0.5kA^2 = 0.5*80*0.2^2 = 1.6 J

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