= 6) Consider an LC circuit with C : 1 F and L = 2 mH. At a given instant, the capacitor has 1/4 of its maximum energy stored, and the inductor is supplying 0.25 W of power to the circuit. What is the maximum charge of the capacitor? 1.6 A body of uniform cross-sectional area A and mass density @ floats in a liquid of density 00 (where 0 < 00), and at equilibrium displaces a volume V. Making use of Archimedes' principle (that the buoyancy force acting on a partially submerged body is equal to the weight of the displaced liquid), show that the period of small amplitude oscillations about the equilibrium position is = 2 V gA 199 1.9 (a) Imagine a straight tunnel passing through the center of the Earth, which is regarded as a sphere of radius R and uniform mass density. A particle is dropped into the tunnel from the surface. Show that the particle undergoes simple harmonic oscillation at the angular frequency w = g/R, where g is the gravitational acceleration at Earth's surface. (Hint: The gravitational acceleration at a point inside a spherically symmetric mass distribution is the same as if all of the mass interior to the point were concentrated at the center, and all of the mass exterior to the point were neglected.) Estimate how long it takes the particle to reach the other end of the tunnel. (b) Assuming that the tunnel is smooth (i.e., ignoring friction), show that motion is simple harmonic even if the tunnel does not pass through the center of the Earth, and that the travel time from one end of the tunnel to the other is the same as before. [From Ingard 1988.]