A 3.50 kg object is attached to a spring and placed on a horizontal, smooth surf

Question

A3.50kg object is attached to a spring and placed on a horizontal, smooth surface. A horizontal force of25.0N is required to hold the object at rest when it is pulled 0.200 m from its equilibrium position (the origin of thexaxis). The object is now released from rest with an initial position ofxi= 0.200 m, and it subsequently undergoes simple harmonic oscillations.

Explanation / Answer

(a) By F = -kx [-ve is just indicating the direction of Force] =>23 = k x (0.2)^2 =>k = 575 N/m (b) By f = 1/2? x ?[k/m] =>f = 1/(2 x 3.14) x ?[575/1] =>f = 3.82 Hz (c) By PE(spring) = KE(spring) at the equilibrium position i.e. at the +0.20 m from the origion =>1/2mv^2 = 1/2kx^2 =>v^2 = 575 x (0.20)^2 =>v = ?23 =>v(max) = 4.80 m/s (d) a(max) = -A?^2 [-ve just indicating the direction]at origin & +0.40 m position =>a(max) = 0.2 x (2 x ? x f)^2 =>a(max) = 0.2 x (2 x 3.14 x 3.82)^2 =>a(max) = 115.10 m/s^2 (e) By PE(spring)[max] = 1/2kx^2 =>U = 1/2 x 575 x (0.20)^2 =>U = 11.50 J (f) By U = 1/2mv^2 + 1/2kx^2 =>11.5 = 1/2 x 1 x v^2 + 1/2 x 575 x (0.2/3)^2 =>v = ?20.44 =>v = 4.52 m/s (g) By a = -A?^2 =>a = -(0.2/3) x (2 x ? x f)^2 =>a = -38.37 m/s^2

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