A 3.30-kg object is attached to a spring and placed on frictionless, horizontal

Question

A 3.30-kg object is attached to a spring and placed on frictionless, horizontal surface. A horizontal force of 23.0 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 =

Explanation / Answer

a) F = kx

23 = k(0.2)

k = 115 N/m

b) w = sqrt(k/m) = sqrt(115/3.30) = 5.90 Hz

c) V_max = Aw = 0.2 x 5.90 = 1.18 m/s

d) at equilibrium position of spring .

x = 0

e) a+_max = Aw^2 = 0.2 x 5.90^2 = 6.96 m/s2

f) at xtreme positions

say at x = +0.2m and -0.2 m

g) total energy of any SHM = kA^2 /2 = 115 x 0.2^2 /2 = 2.3 J

h) x = Acos(wt)

A/3 = Acos(wt)

cos(wt) = 1/3

wt = cos-1(1/3) = 0.39 pi

t = 0.39 x 3.41 / 5.90 = 0.21 sec

v = V_max sin(wt) = 1.18 sin(5.90 x0.21)

= 1.15 m/s

i) a = a_max cos(wt)

a = 6.96 cos(5.90 x0.21)

a = 2.32 m/s2

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