Damping

Friction, and the three ways a system can settle.

12 problems · about 24 minutes · F-IF.B.4, A-REI.B.4

What this lesson teaches

The student classifies damped systems as under, critically or overdamped.

Damping adds a term proportional to velocity, and the discriminant c² − 4mk decides everything.
Negative means underdamped and oscillating; positive means overdamped and sluggish.
Zero is critical damping, which returns to rest fastest without overshooting.

Straightforward practice. Get the method working first.

5 problems

m = 1, c = 4, k = 4. What is c² − 4mk?
Answer 0
Why 0, which is critical damping.
Which damping case oscillates?
Answer Underdamped, where the roots are complex.
Why Only the underdamped case.
m = 1, c = 2, k = 5. What is c² − 4mk?
Answer -16
Why 4 − 20.
A discriminant of −16. Enter 1 underdamped, 2 critically damped, 3 overdamped.
Answer 1
Why Negative means complex roots.
A discriminant of 0. Enter 1 underdamped, 2 critically damped, 3 overdamped.
Answer 2
Why Exactly on the boundary.

The same ideas with more to keep track of.

3 problems

A discriminant of 25. Enter 1 underdamped, 2 critically damped, 3 overdamped.
Answer 3
Why Positive means two real roots.
m = 1 and k = 16. What value of c gives critical damping?
Answer 8
Why c² = 64.
Which case returns to rest fastest without overshooting? Enter 1 under, 2 critical, 3 over.
Answer 2
Why That is what critical means.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Match each case to how the system behaves.
Answer Underdamped → Oscillates inside a shrinking envelope; Critically damped → Returns to rest fastest without overshooting; Overdamped → Crawls back slowly with no oscillation
Why Only one of them oscillates.
c = 0, meaning no damping at all. Does the system oscillate forever? 1 yes, 0 no.
Answer 1
Why Nothing removes energy.
The Discriminant: m = 1, c = 6, k = 5. What is c² − 4mk?
Answer 16
Why 16.
The Boundary: m = 1 and k = 9. What value of c gives critical damping?
Answer 6
Why 6.