I7-I8 Simple harmonic motion and damped oscillations Quiz: AQA Further Maths, Unit 2

The 20 questions

1. Which differential equation describes simple harmonic motion?

   • x' = -w x
   • x'' = -w x
   • x'' = -w^2 x
   • x'' = w^2 x

2. For simple harmonic motion x'' = -w^2 x, what is the period?

   • 2 pi / w
   • w / (2 pi)
   • pi / w
   • 2 pi w

3. If x = A cos(wt) + B sin(wt), what is the amplitude of the motion?

   • A + B
   • A^2 + B^2
   • |A - B|
   • sqrt(A^2 + B^2)

4. For a mass m on a spring of stiffness k, what is the angular frequency w of the simple harmonic motion?

   • k/m
   • sqrt(k/m)
   • mk
   • sqrt(m/k)

5. For x'' + 2 lambda x' + w^2 x = 0, which condition gives light damping?

   • lambda = 0
   • lambda > w
   • lambda < w
   • lambda = w

6. What condition defines critical damping for x'' + 2 lambda x' + w^2 x = 0?

   • lambda = w
   • lambda > w
   • lambda < w
   • w = 0

7. What happens to the motion in heavy damping, lambda > w?

   • The auxiliary equation has a repeated root
   • The auxiliary equation has no real solution
   • The motion oscillates with complex roots
   • The motion is non-oscillatory with two distinct real roots

8. The discriminant of m^2 + 2 lambda m + w^2 = 0 is which expression?

   • 4(lambda^2 - w^2)
   • 4(w^2 - lambda^2)
   • lambda^2 + w^2
   • 2 lambda - w^2

9. What is the period of motion for x'' = -16x?

   • 4 pi
   • pi/2
   • pi
   • 2 pi

10. Solve x'' + 4x = 0 with x(0) = 2 and x'(0) = 0.

    • x = 4 cos(2t)
    • x = 2 cos(2t)
    • x = 2 cos(4t)
    • x = 2 sin(2t)

11. A mass of 2 kg is attached to a spring with stiffness 8 N/m. What is the period of the motion in seconds?

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

12. Which type of damping does x'' + 6x' + 9x = 0 represent?

    • Critical damping
    • Light damping
    • Heavy damping
    • Undamped motion

13. Which expression is the general solution of x'' + 2x' + 5x = 0?

    • e^(t)(A cos(2t) + B sin(2t))
    • e^(-t)(A cos(2t) + B sin(2t))
    • e^(-2t)(A cos(t) + B sin(t))
    • A e^(-t) + B e^(-5t)

14. Which expression is the general solution of x'' + 5x' + 4x = 0?

    • (A + Bt) e^(-t)
    • e^(-t)(A cos(4t) + B sin(4t))
    • A e^(-5t) + B e^(-4t)
    • A e^(-t) + B e^(-4t)

15. Solve x'' + 2x' + x = 0 with x(0) = 1 and x'(0) = 0.

    • x = (1 - t) e^(-t)
    • x = (1 + 2t) e^(-t)
    • x = e^(-t)
    • x = (1 + t) e^(-t)

16. The displacement is x = 3 cos(4t) - 4 sin(4t). What is the amplitude?

    • 1
    • 5
    • sqrt(7)
    • 7

17. What are the roots of m^2 + 4m + 1 = 0?

    • m = 2 +/- sqrt(3)
    • m = -2 +/- sqrt(5)
    • m = -2 +/- sqrt(3)
    • m = -4 +/- 1

18. For x'' + 4x' + 9x = 0, which type of damping occurs?

    • Critical
    • Light (underdamped)
    • No damping
    • Heavy

19. Solve x'' + 4x' + 4x = 0 with x(0) = 0 and x'(0) = 1.

    • x = t e^(-2t)
    • x = 2t e^(-2t)
    • x = e^(-2t)
    • x = t e^(-t)

20. For x = e^(-t)(A cos(2t) + B sin(2t)), what is the limit of x as t tends to infinity?

    • 0
    • 1
    • A
    • infinity

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