StudyBrixFree worksheets, study guides and courses

Physics · lesson 2 of 4

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.

University8 practice questionsNotes + worked examples
Take the quizStudy guideRevision sheet

Learn the ideas

Prerequisites: Second-order differential equations and complex numbers.

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.

Worked examples

Example 1

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

ω0=sqrt(k/m)=sqrt(9).

3 rad/s

Example 2

Which force represents linear viscous damping?

The force opposes velocity and removes mechanical energy.

−b times velocity with b positive

Check your understanding

8 questions

Decimals or fractions are accepted for numeric answers. Proofs and written responses use sample answers and require human marking. This is practice, not a protected examination.

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

Show answer and explanation

3 rad/s

ω0=sqrt(k/m)=sqrt(9).

2 Which force represents linear viscous damping?

Show answer and explanation

−b times velocity with b positive

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.s

Show answer and explanation

2 s

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

4 What distinguishes a transient from a steady driven response?

Show answer and explanation

A transient depends on initial conditions and may decay

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 and marking guidance

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

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?

Show answer and explanation

2

ω0 scales as sqrt(k).

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

Show answer and explanation

−ω²x

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 and 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.

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

Further reading and review references

These are reference links, not copied exercises, institutional endorsement or proof of independent review.