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# Simple harmonic oscillation driven by an external force

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2. A spring with a 4-kg mass has natural length 1 m and is maintained stretched to a length of 1.3 m by a force of 24.3 N. If the spring is compressed to a length of 0.8 m and then released with zero velocity, find the position of the mass at any time t.

10. As in Exercise 9, consider a spring with mass m, spring constant k, and damping constant c = 0, and let w = sqrt (k/m). If an external force F(t) = F0 cos wt is applied (the applied frequency equals the natural frequency), use the method of undetermined coefficients to show that the motion of the mass is given by x(t) = c1 cos wt + c2 sin wt + (F0/(2mw))t sin wt.

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#### Solution Preview

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First we need to find the spring constant. Assume the equilibrium (no compress or stretch in the spring) is x=0.
When the spring is stretched to a length of 1.3 m, the displacement is x = 1.3- 1 = 0.3 m.

So the spring constant is .

Now based on the Newton's second law,

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#### Solution Summary

By solving differential equations, It finds the motion equations of the mass, which is in simple harmonic oscillation driven by an external force. The solution is detailed and well presented.

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