Physics · University
Underdamped envelope after a time
This is the amplitude envelope, not instantaneous signed displacement. The oscillator is specified as underdamped.
Method and conditions
This is the amplitude envelope, not instantaneous signed displacement. The oscillator is specified as underdamped.
A(t) = A₀ exp[−bt/(2m)]
Worked examples
An underdamped oscillator obeys m x″+b x′+kx=0 with m=7 kg, b=0.2 kg/s and initial envelope 39 cm. Find its envelope after 4 s. Round the final result to 4 decimal places if needed. Answer in cm.
The decay constant is b/(2m) = 0.2/(14) s⁻¹. Multiply the initial envelope by exp[−bt/(2m)].
36.8339 cm
An underdamped oscillator obeys m x″+b x′+kx=0 with m=2 kg, b=0.1 kg/s and initial envelope 24 cm. Find its envelope after 20 s. Round the final result to 4 decimal places if needed. Answer in cm.
The decay constant is b/(2m) = 0.1/(4) s⁻¹. Multiply the initial envelope by exp[−bt/(2m)].
14.5567 cm
Try it
An underdamped oscillator obeys m x″+b x′+kx=0 with m=6 kg, b=0.8 kg/s and initial envelope 30 cm. Find its envelope after 12 s. Round the final result to 4 decimal places if needed. Answer in cm.
An underdamped oscillator obeys m x″+b x′+kx=0 with m=2 kg, b=0.1 kg/s and initial envelope 14 cm. Find its envelope after 19 s. Round the final result to 4 decimal places if needed. Answer in cm.
Check your work
Answers and marking notes
- 13.4799 cm
The decay constant is b/(2m) = 0.8/(12) s⁻¹. Multiply the initial envelope by exp[−bt/(2m)].
- 8.7064 cm
The decay constant is b/(2m) = 0.1/(4) s⁻¹. Multiply the initial envelope by exp[−bt/(2m)].
Further reading and review references
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