Question

If the differential equationmd2xdt2+3dxdt+7x=0is overdamped, the range of values for m is? ______Your answer will be an interval of numbers given in the form (1,2), [1,2), (-inf,6], etc.

Answer 1

In an overdamped system, the range of values for m is m > 0.

Step-by-step explanation:

If the differential equation md2xdt2 + 3dxdt + 7x = 0 is overdamped, the range of values for m is:

To determine the solution to this equation, consider the plot of position versus time shown in Figure 15.26. The curve resembles a cosine curve oscillating in the envelope of an exponential function e^(-αt) where α = √(3m). The solution is: x(t) = Ae^(-αt) + Be^(αt), where A and B are constants.

Therefore, for an overdamped system, α > 0. Since α = √(3m), we have √(3m) > 0. Solving for m, we find that the range of values for m is: m > 0.

Answer 2

The range of values for m in the given differential equation md^2x/dt^2 + 3(dx/dt) + 7x = 0 that would produce an overdamped system is m < 9/28.

Step-by-step explanation:

An overdamped system moves more slowly toward equilibrium than one that is critically damped. In the given differential equation md2xdt2 + 3dxdt + 7x = 0, the range of values for m that would produce an overdamped system can be determined by comparing the coefficients of the equation.

For an overdamped system, the discriminant of the characteristic equation should be positive. In this case, the discriminant is (3)^2 - 4(m)(7) = 9 -28m. For the discriminant to be positive, we have 9 - 28m > 0. Solving this inequality gives us m < 9/28.