A 1.80-kg object is attached to a spring and placed on frictionless, horizontal

Question

A 1.80-kg object is attached to a spring and placed on frictionless, horizontal surface. A horizontal force of 25.0 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=± (e) Find the maximum acceleration of the object. m/s2 (f) Where does the maximum acceleration occur? (g) Find the total energy of the oscillating system (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) Fnet= k x - F = 0

k (0.200) = 25

k = 125 N/m

(B) f = sqrt(k/m)/2pi = 1.33 Hz

(C) A = 0.20 m

w = 2pi f = 8.33 rad/s

v_max = A w = 1.67 m/s

(D) at x = 0

(E) a_max = w^2 A = 13.9 m/s^2

(f) x = -0.2 and +0.2

(g) ME = 125(0.2^2)/2 = 2.5 J

(h) v^2 = w^2 (A^2 - x^2)

v^2 = 8.22^2 (0.2^2 - (0.2/3)^2)

v = 1.57 m/s

(i) a = (125)(0.2 /3) /(1.80) = 4.63 m/s^2

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