The ODE mx'' + cx' + kx = 0 is used to model a damped mass-spring system, where m is the mass, c is the damping constant and k is the spring constant. Describe and explain the behaviour of the system for the cases: (a) c^2>4mk; (b) c^2=4mk; (c) c^2<4mk.

In the case c2>4mk, the characteristic equation has two distinct real roots; this represents overdamping. The system does not oscillate, and x approaches zero as time approaches infinity.In the case c2=4mk, the characteristic equation has a repeated real root; this represents critical damping. The system does not oscillate and returns to its equilibrium position in the shortest possible time; x approaches zero as time approaches infinity.In the case c2<4mk, the characteristic equation has two complex routes; this represents underdamping. The system oscillates with an exponentially decreasing amplitude; the amplitude of oscillations approaches zero as time approaches infinity.

Related Further Mathematics A Level answers

All answers ▸

Let E be an ellipse with equation (x/3)^2 + (y/4)^2 = 1. Find the equation of the tangent to E at the point P where x = √3 and y > 0, in the form ax + by = c, where a, b and c are rational.


Prove by induction that, for all integers n >=1 , ∑(from r=1 to n) r(2r−1)(3r−1)=(n/6)(n+1)(9n^2 -n−2). Assume that 9(k+1)^2 -(k+1)-2=9k^2 +17k+6


solve the 1st order differential equation 2y+(x*dy/dx)=x^3


Are the integers a group under addition? How about multiplication?


We're here to help

contact us iconContact usWhatsapp logoMessage us on Whatsapptelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences