shm assessed hw markscheme

1.       (a)     The resultant force is zero (WTTE)

For the first mark allow
-      sum of forces is zero,
-      upward force = downward force,
-      forces cancel each other
BUT do not allow forces are balanced

B1

          Forces are weight and force from the spring (allow tension)

Allow force of gravity for weight

B1

  (b)   (i)      acceleration is (directly) proportional to displacement

M1

          and is directed in the opposite direction to the displacement. (WTTE)

allow a = –(2πf)2 x, provided a and x are identified and –ve sign must be explained.
Do not allow “acceleration is prop to negative displacement for second mark.
Allow always towards the equilibrium position

A1

  (ii)   x = acos2πft  2πf = 7.85 (expressed in any form)

M1

          f = (7.85/2π) = 1.25 (1.249Hz)

Do not allow use of the fig to show T = 0.8s and hence
f = 1.25 Hz. This scores 0.

A1

  (iii)  correct substn in Vmax = (2πf)A  Vmax = 2π × 1.25 × 0.012

Many will forget to change 12 mm into 0.012m and have
v = 94 m s–1 this scores 1 mark.

C1

          Vmax = 0.094 m s–1

A1

  (c)   roughly sinusoidal graph of correct period ie 0.8s

B1

          90° out of phase with displacement graph (i.e. starts at origin
with -ve initial gradient)

B1

          maximum velocity correctly shown as 0.094 {allow ecf from (iii)}

B1

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  2.     (i)      1.      Measure the time t for N oscillations.                                                   M1
frequency f = N/t                                                                                  A1

2.       Measure the amplitude A of the oscillations using the ruler.                 M1
maximum speed is calculated using: vmax = (2πf)A                               A1

  (ii)   The maximum speed is doubled                                                                    B1
because the frequency is the same and vmax =  2 pi fA

(iii)    F = (–) kx and F = ma
Therefore ma = (–) kx                                                                                   M2
ω2 = k / m
T = 2Pi root k over m                                                                                              M1

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  3.     (a)     (i)      A motion in which the acceleration/force is proportional to the
displacement; (1)
directed towards the centre of oscillation/equilibrium position/AW
or a α -x or a = –ω2x or a = –4π2f2x; symbols must be identified (1)     2

  (ii)   T = 0.25 s or f = 1/T; f = 4 (Hz) (2)                                                         2

(iii)    a = –4π2f2A; = 4 × 9.87 × 16 × 0.005; = 3.2 (m s–2) ecf a(ii) (3)             3

  (b)   (i)      Resonance occurs at /close to the natural frequency of an oscillating (1)
object/system; caused by driving force (at this frequency); when (1)
maximum energy transfer between driver and driven/maximum
amplitude achieved (1)                                                                            3

3 marking points in any sensible order

  (ii)   1   reduced amplitude; as resonance frequency lower
     or resonance will occur at lower frequency; as greater
     inertia/reduced natural frequency/AW in terms of amplitude change (2)

          2   reduced amplitude; as resonance frequency higher
     or resonance will occur at a higher frequency; as larger restoring
     force/increased natural frequency/AW in terms of amplitude change (2)           4

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  4.     (a)     (i)      acceleration ∞ displacement; indication of restoring force by negative
sign/acc. in opp. direction to displacement/acc. towards origin/AW        2

  (ii)   linear graph through origin; negative gradient                                          2

  (b)   (i)      0.05 (m)                                                                                                  1

  (ii)   4π2f2 = a/A; = 12.5/0.05 = 250 so f = 2.5(1) Hz; T = 1/f (= 0.4 s)           3

  (c)    (i)      cosine wave; correct period of 0.4 s; correct amplitude of 0.05 m          3

  (ii)   0; 0.1/0.3/0.5/0.7/0.9 (s)                                                                          2

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