# Driven oscillations, damping and resonance

> A linear mass-spring oscillator obeys mx″+kx=0. Its natural angular frequency is sqrt(k/m), not k/m.

Course: [University mechanics and waves](https://studybrix.com/en/courses/university-mechanics/index.md)
Level: university
Prerequisites: Second-order differential equations and complex numbers.

## Notes

### Free motion

A linear mass-spring oscillator obeys mx″+kx=0. Its natural angular frequency is sqrt(k/m), not k/m.

### Damping

A viscous damping force −bx′ converts the equation to mx″+bx′+kx=0. In underdamping, the oscillation envelope decays exponentially.

### Driving

A periodic external force can transfer energy to the oscillator. The steady response depends on drive frequency, damping and phase.

### Resonance

A finite drive does not imply infinite amplitude in a damped physical oscillator. Transient motion and nonlinear limits matter.

## Practice questions

### 1. For m=2 kg and k=18 N/m, find the undamped angular frequency.

Answer: 3 rad/s

Explanation: ω0=sqrt(k/m)=sqrt(9).

### 2. Which force represents linear viscous damping?

A. Mass times position
B. −b times velocity with b positive
C. A constant upward force only
D. +b times displacement

Answer: −b times velocity with b positive

Explanation: The force opposes velocity and removes mechanical energy.

### 3. An underdamped amplitude follows A(t)=8 exp(−t/2) cm. Find the exponential decay time constant.

Answer: 2 s

Explanation: Write the envelope as A0 exp(−t/τ); τ=2 s.

### 4. What distinguishes a transient from a steady driven response?

A. A transient must last forever
B. A steady response has no frequency
C. A transient depends on initial conditions and may decay
D. A steady response cannot lose energy

Answer: A transient depends on initial conditions and may decay

Explanation: Damping can remove the initial-condition contribution while driving maintains a steady oscillation.

### 5. Why does energy not grow without limit in an ordinary damped oscillator driven near resonance?

Sample answer / marking guidance: In steady state, average driving power balances dissipated power. Damping limits the linear response, and large-amplitude nonlinearities can further change it.

Explanation: Credit energy balance and avoidance of an undamped infinite-amplitude generalization.

### 6. If k is multiplied by 4 at fixed mass, by what factor does ω0 change?

Answer: 2

Explanation: ω0 scales as sqrt(k).

### 7. A displacement is x=A cos(ωt). What is its acceleration?

A. +ωx
B. A constant A
C. Zero at all times
D. −ω²x

Answer: −ω²x

Explanation: Two derivatives give x″=−ω²A cos(ωt).

### 8. Describe an experiment using a recorded decay envelope to estimate damping, without prescribing a hazardous apparatus.

Sample answer / marking guidance: Plot ln of successive amplitude magnitudes against time in the linear underdamped regime. A straight-line slope estimates −b/(2m), subject to measurement uncertainty.

Explanation: Credit use of the envelope, logarithm and linear-model assumptions.

## Further reading

- https://ocw.mit.edu/courses/8-09-classical-mechanics-iii-fall-2014/
- https://openstax.org/details/books/university-physics-volume-3

Original AI-assisted educational draft; not independently teacher-reviewed. References support further study, not copied exercises, affiliation or curriculum accreditation.