91Ó°ÊÓ

\(x^{\prime \prime}+64 x=0, x(0)=3 / 4, x^{\prime}(0)=-2\)

. If \(L=1 \mathrm{~m}\), how many ticks does the clock make in \(1 \mathrm{~min}\) if it ticks once for each time the pendulum makes a complete swing?

A 6 -lb object stretches a spring \(6 \mathrm{in}\). If the object is lifted 3 in above the equilibrium position and released, determine the time required for the object to return to its equilibrium position. What is the displacement of the object at \(t=5 \mathrm{~s}\) ? If the object is released from its equilibrium position with a downward initial velocity of \(1 \mathrm{ft} / \mathrm{s}\), determine the time required for the object to return to its equilibrium position.

\(2 d^{2} x / d t^{2}+4 d x / d t+x=0, x(0)=-1\), \(x^{\prime}(0)=2\)

(Archimedes' principle) Suppose that an object of mass \(m\) is submerged (either partially or totally) in a liquid of density \(\rho\). Archimedes' principle states that a body in liquid experiences a buoyant upward force equal to the weight of the liquid displaced by the body. The object is in equilibrium when the buoyant force of the displaced liquid equals the force of gravity on the object. Consider the cylinder of radius \(r\) and height \(H\) of which \(h\) units of the height is submerged at equilibrium, as indicated in the following figure. (a) Show that the weight of liquid displaced at equilibrium is \(\pi r^{2} h \rho\). Therefore, at equilibrium, \(\pi r^{2} h \rho=m g\). (b) Let \(y=y(t)\) represent the vertical displacement of the cylinder from equilibrium. Show that when the cylinder is raised out of the liquid, the downward force is \(\pi r^{2}(h-\) y) \(\rho\). Use Newton's second law of motion to show that \(m y^{\prime \prime}=\pi r^{2}(h-y) \rho-m g\). Simplify this equation to obtain a second order equation that models this situation.