Question

I AM REALLY STRUGGLING ON THIS PROBLEM PLEASE HELP ME CORRECT AND NEAT WORK IS MUCH APPRECIATED THANKS

(1 point) If the differential equation m dx +4 dt + 7x = 0 dt2 is 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

The differential equation of a damped harmonic oscillation is given as:

m dc + b dt + kr = 0 dt2

where b is called the damping constant, and the oscillation is overdamped, underdamped, or critically damped according to:

• Underdamped
  
  $\small b^2<4km$
  
  
• Critically damped
  
  $\small b^2=4km$
  
  
• Overdamped
  
  $\small b^2>4km$

Now, we can use the above conditions to solve our problem. We are given the equation

$\small m{d^2x\over dt^2}+4{dx\over dt}+7x=0$

Thus, here b = 4, k = 7

Now, the motion is given to be overdamped, hence we must have

$\small b^2>4km \Rightarrow 16 > 7m \Rightarrow m < {16\over 7}$

Thus, the range of values of m for which the motion will be underdamped is

$\small \left (-\infty, {16\over 7}\right )$

write (-inf,16/7) in the answering box