Chapter 5: 5RP (page 196)
In the presence of a damping force, the displacements of a mass on a spring will always approach zero as __________
The given Statement is “ In the presence of a damping force, the displacements of a mass on a spring will always approach zero as “ that is False.
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Suppose that a uniform thin elastic column is hinged at the endand embedded at the end
(a) Use the fourth-order differential equation given in Problem 25 to find the eigenvaluesthe critical loadsthe Euler loadand the deflections
(b) Use a graphing utility to graph the first buckling mode.
Suppose a pendulum is formed by attaching a massto theend of a string of negligible mass and length l. Atthependulum is released from rest at a small displacement angleto the right of the vertical equilibrium position OP. SeeFigure 5.R.5. At timethe string hits a nail at a point N onOP a distancefrom O, but the mass continues to the left asshown in the figure.
(a) Construct and solve a linear initial-value problem for thedisplacement angleshown in the figure. Find theintervalon whichis defined.
(b) Construct and solve a linear initial-value problem for thedisplacement angle shown in the figure. Find theintervalon which is defined, where isthe time that m returns to the vertical line NP.
Constant-Harvest Model A model that describes the population of a fishery in which harvesting takes place at a constant rate is given by
where k and h are positive constants.
(a) Solve the DE subject to .
(b) Describe the behavior of the population P(t) forincreasing_time in the three cases
(c) Use the results from part (b) to determine whether the fish population will ever go extinct in finite time, that is, whether there exists a time
such that . If the population goes extinct, then find T.
Use a root-finding application of a CAS to approximate the first four eigenvalues and for the BVP in Problem 38 .
Rotating of a Shaft
Suppose the x-axis on the intervalis the geometric center of a long straight shaft, such as the propeller shaft of a ship. See Figure 5.2.12. When the shaft is rotating at a constant angular speed about this axis the deflectionof the shaft satisfies the differential equation
Where is its density per unit length. If the shaft is simplify supported, or hinged , at both ends the boundary conditions are then,
(a) If , then find the eigenvalues and eigenfunctions for this boundary – value problem.
(b) Use the eigenvalues in part (a) to find corresponding angular speeds . The values are called critical speeds. The value is called the fundamental critical speed and analogous to example 4, at this speed the shaft changes shape from to a deflection given by .
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