A 3. horizontal force of 20.0 N is required to hold the mass at rest when it is

Question

A 3. horizontal force of 20.0 N is required to hold the mass at rest when it is pulled 0.200 m from its equilibrium position (the origin of the x axis). The mass is now released from rest with an initi displacement of x 200 m, and it subsequently undergoes simple harmonic oscillations. Find: a) the force constant of the spring, A 2.0 kg mass is attached to a spring and placed on a horizontal, frictionless surface. al b) the frequency of oscillation, c) the maximum speed of the mass. Where does this maximum occur? d) the maximum acceleration of the mass. Where does it occur? e) the total energy of the oscillating system, D the speed when displacement equals one-third of the maximum value, g) the acceleration when displacement equals one-third of the maximum.

Explanation / Answer

3. (A) F = k x

20 = k (0.200)

k = 100 N/m

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

f = sqrt(100 / 2) / 2pi

f = 1.125 Hz  

(C) w = sqrt(k/m) = 7.071 rad/s

A = 0.2 m

v_max = A w = 1.414 m/s

at x = 0

(D) a_max = A w^2 = 10 m/s^2

at x = - 0.2 and 0.2 m

(E) total energy = k A^2 / 2 = 100 x 0.2^2 / 2

= 2 J

(F) Pe = 100(0.2/3)^2 / 2 = 0.22 J

KE = 2 - 0.22 = 1.78 J

2 v^2 / 2 = 1.78

v = 1.33 m/s

(g) a = k x / m = 100(0.2/3)/2 = 3.33 m/s^2

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