9. A spring-mass system exhibits free, undamped motion and its position x() is depicted graphically below, where x() is measured in feet and t is measured in seconds. Assume that the mass weighs 8 pounds. Note: The horizontal axis is t and the vertical axis is x(t). Remember that positions below the equilibrium position are considered to be positive. a) Express the motion of the mass in the form x(t) =c₁cos(wt) + c2sin(wt). Give exact values for all quantities. x(t) т ° B t b) What is the initial velocity of the mass? (Exact or a 1 decimal place value is allowed here) c) What is the exact value of k, the spring constant for the system?

The density of air changes with height. Under some conditions density p, depends on height z, and temperature T according to the following equation where Po and A are both constants. A meteorological balloon ascends (i.e., starts at z = 1 and gains height) over the course of several hours. Complete parts (a) and (b) below. Az P(z,T) = Po e ..... dz v) and that the temperature changes over time (i.e., that T is given by a function T(t)), derive, using the chain rule, an expression for the rate of change of air density, (a) Assuming that the balloon ascends at a speed v (i.e., dt as measured by the weather balloon. Choose the correct answer below. dp Az dT O A. dt T2 dt dp %3D dt Az dT dp С. dt %3D + T dt dp Az) dT O D. dt T2) dt (b) Assume that v = 1, Po = 1, and A = 1 and that when t= 0, T= 1. Are there any conditions under which the density, as measured by the balloon will not change in time? That is, find a differential equation that T must satisfy, if dp = 0, and solve this…

A mass weighing 20 pounds stretches a spring 6 inches. The mass is initially released from rest from a point 8 inches below the equilibrium position. (a) Find the position x of the mass at the times  t = ?/12, ?/8, ?/6, ?/4, and 9?/32 s.  (Use  g = 32 ft/s2  for the acceleration due to gravity.)