Question

Consider a spring mass system with mass 0.5 kg and spring constant 𝑘 = 2 Nm^-1 in a viscous medium with drag coefficient 𝑏 = 3 kg s ^-1 . The additional mass required so that the motion becomes critically damped is _______ kg (rounded off to three decimal places)

Answer 1

Correct Answer: 0.62 - 0.63

Given:
Mass \(m=0.5\ \text{kg}\), spring constant \(k=2\ \text{N\,m}^{-1}\), damping (drag) coefficient \(b=3\ \text{kg\,s}^{-1}\).

For critical damping: 
The equation is \(m_{\rm total}\,x'' + b\,x' + kx = 0\). Critical damping occurs when \[ b = 2\sqrt{k\,m_{\rm total}}. \] Solve for the required total mass \(m_{\rm total}\): \[ m_{\rm total}=\frac{b^2}{4k}. \] Compute \(m_{\rm total}\): \[ m_{\rm total}=\frac{3^2}{4\times 2}=\frac{9}{8}=1.125\ \text{kg}. \] Additional mass required:
\[ \Delta m = m_{\rm total}-m = 1.125-0.5 = 0.625\ \text{kg}. \] Final answer (rounded to three decimals):
\[ \boxed{\Delta m = 0.625\ \text{kg}} \]