3.10E: Exercises for Section 3.10

Exercise \(\PageIndex{1}\)

Derive a formula for \(x_{sp}\) if the equation is \( mx'' + cx' + kx = F_0 \sin ( \omega t) \). Assume \( c > 0 \).

Exercise \(\PageIndex{2}\)

Derive a formula for \( x_{sp}\) if the equation is \( mx'' + cx' + kx = F_0 \cos (\omega t) + F_1 \cos (3 \omega t)\). Assume \( c > 0 \).

Exercise \(\PageIndex{3}\)

Take \( mx'' + cx' + kx = F_0 \cos ( \omega t) \). Fix \(m > 0\) and \( k > 0\). Now think of the function \( C(\omega ) \). For what values of \(c\) (solve in terms of \( m, k,\) and \(F_0\) ) will there be no practical resonance (that is, for what values of \(c\) is there no maximum of \(C(\omega )\) for \(\omega > 0 \) )?

Exercise \(\PageIndex{4}\)

Take \(mx'' + cx' + kx = F_0 \cos (\omega t) \). Fix \( c > 0\) and \(k > 0 \). Now think of the function \( C(\omega ) \). For what values of \(m\) (solve in terms of \(c, k, \) and \(F_0 \)) will there be no practical resonance (that is, for what values of \(m\) is there no maximum of \(C(\omega )\) for \(\omega > 0\) )?

Exercise \(\PageIndex{5}\)

Suppose a water tower in an earthquake acts as a mass-spring system. Assume that the container on top is full and the water does not move around. The container then acts as a mass and the support acts as the spring, where the induced vibrations are horizontal. Suppose that the container with water has a mass of \(m = 10, 000 kg \). It takes a force of \(1000\) newtons to displace the container \(1\) meter. For simplicity assume no friction. When the earthquake hits the water tower is at rest (it is not moving).

Suppose that an earthquake induces an external force \( F(t) = mA \omega^2 \cos (\omega t). \)

1. What is the natural frequency of the water tower?
2. If \(\omega \) is not the natural frequency, find a formula for the maximal amplitude of the resulting oscillations of the water container (the maximal deviation from the rest position). The motion will be a high frequency wave modulated by a low frequency wave, so simply find the constant in front of the sines.
3. Suppose \(A = 1\) and an earthquake with frequency \(0.5\) cycles per second comes. What is the amplitude of the oscillations? Suppose that if the water tower moves more than \(1.5\) meter, the tower collapses. Will the tower collapse?

Exercise \(\PageIndex{6}\)

A mass of \(4\text{ kg}\) on a spring with \(k = 4\) and a damping constant \(c = 1\). Suppose that \(F_0 = 2 \). Using forcing of \( F_0 \cos (\omega t) \). Find the \(\omega \) that causes practical resonance and find the amplitude.

Answer

\(\omega=\frac{\sqrt{31}}{4\sqrt{2}}\approx 0.984\quad C(\omega )=\frac{16}{3\sqrt{7}}\approx 2.016\)

Exercise \(\PageIndex{7}\)

Derive a formula for \(x_{sp}\) for \( mx'' + cx' + kx = F_0 \cos (\omega t) + A \) where \(A\) is some constant. Assume \( c > 0 \).

Answer

\(x_{sp}=\frac{(\omega_{0}^{2}-\omega^{2})F_{0}}{m(2\omega p)^{2}+m(\omega_{0}^{2}-\omega^{2})^{2}}\cos (\omega t)+\frac{2\omega pF_{0}}{m(2\omega p)^{2}+m(\omega_{0}^{2}-\omega^{2})^{2}}\sin (\omega t)+\frac{A}{k}\), where \(p=\frac{c}{2m}\) and \(\omega_{0}=\sqrt{\frac{k}{m}}\)

Exercise \(\PageIndex{8}\)

Suppose there is no damping in a mass and spring system with \( m = 5, k = 20, \) and \(F_0 = 5 \). Suppose that \( \omega \) is chosen to be precisely the resonance frequency.

1. Find \(\omega \).
2. Find the amplitude of the oscillations at time \(t = 100\).

Answer

1. \(\omega =2\)
2. \(25\)