9.8 Modelling damped harmonic motion Quiz: Pearson Edexcel Further Maths, Unit 9

The 20 questions

1. Which equation models damped oscillation of a mass m with resistance proportional to speed, spring constant k and damping coefficient c?

   • m x'' + k x = 0
   • m x' + c x + k = 0
   • m x'' - c x' + k x = 0
   • m x'' + c x' + k x = 0

2. In damped harmonic motion, what is the resistance force proportional to?

   • The velocity
   • The displacement
   • The square of the time
   • The acceleration

3. Which auxiliary equation case gives underdamped (oscillatory, decaying) motion?

   • A repeated real root
   • Two distinct negative real roots
   • A zero root
   • Complex roots

4. What characterises overdamped motion?

   • Constant amplitude oscillation
   • Two distinct negative real roots and no oscillation
   • A repeated root giving a single oscillation
   • Complex roots with a decaying sine term

5. Which auxiliary equation case corresponds to critical damping?

   • A zero root
   • A repeated real root
   • Complex roots
   • Two distinct positive roots

6. For x'' + 2x' + 2x = 0, the amplitude envelope is proportional to which expression?

   • e^(-2t)
   • e^(-t)
   • e^t
   • 1/t

7. For x'' + 6x' + 5x = 0, what type of damping occurs?

   • Undamped
   • Underdamped
   • Critically damped
   • Overdamped

8. For x'' + 4x' + 4x = 0, what type of damping occurs?

   • Overdamped, with two distinct roots
   • Critically damped
   • Undamped, with pure oscillation
   • Underdamped, with complex roots

9. For x'' + 2x' + 10x = 0, what is the angular frequency of the oscillation?

   • sqrt(10)
   • 10
   • 1
   • 3

10. Given x'' + 2x' + 5x = 0 with x(0) = 1 and x'(0) = 0, find the constant B in x = e^(-t)(cos 2t + B sin 2t).

    • 2
    • 1
    • 1/2
    • -1/2

11. For a lightly damped oscillator x = e^(-t)(A cos t + B sin t), how long does the amplitude take to fall to e^(-2) of its initial value?

    • 1
    • 4
    • 2
    • ln 2

12. For x'' + 4x' + 3x = 0 with x(0) = 0 and x'(0) = 1, what is the constant A in x = A e^(-t) + B e^(-3t)?

    • -1/2
    • 1
    • 3
    • 1/2

13. What does the particular integral represent in forced damped motion?

    • The steady-state long-term response
    • The initial transient motion
    • The total energy of the system
    • The natural frequency of the system

14. For x'' + 2x' + 2x = 0, what is the period of the damped oscillation?

    • 2 pi
    • 4 pi
    • pi/2
    • pi

15. For x'' + 2x' + 10x = 0, after what time does the amplitude fall to 10% of its initial value?

    • ln 2, about 0.69
    • ln 10, about 2.30
    • 0.1
    • 10

16. A damped oscillator has amplitude proportional to e^(-t). After what time does the amplitude halve?

    • ln 2
    • 2 ln 2
    • 1/2
    • 1

17. Why does a lightly damped system oscillate?

    • Its auxiliary roots are zero, so the motion is constant
    • Its auxiliary roots are complex, so the solution contains a sine or cosine term
    • Its damping coefficient is zero, so the energy is constant
    • Its auxiliary roots are equal, so the solution is a straight line

18. Which statement describes forced vibration in this topic?

    • The spring constant is changed to zero, so the system no longer oscillates
    • A time-dependent external force is added to the right-hand side of the equation
    • A damping term is multiplied by the displacement, so the equation changes form
    • The initial displacement is changed while the equation stays homogeneous

19. For x'' + 5x' + 4x = 0, which describes the long-term behaviour?

    • The displacement oscillates with constant amplitude
    • The displacement tends to the value 4
    • The displacement tends to zero without oscillating
    • The displacement grows without bound

20. The discriminant of x'' + ax' + bx = 0 with a = 4 and b = 3 is 4. What type of damping is this?

    • Overdamped, since the discriminant is positive
    • Overdamped, since the discriminant is 12
    • Critically damped, since the discriminant is zero
    • Underdamped, since the discriminant is negative

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