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Date May 2021 Marks available 2 Reference code 21M.2.HL.TZ1.8
Level Higher level Paper Paper 2 Time zone 1
Command term Estimate Question number 8 Adapted from N/A

Question

On a guitar, the strings played vibrate between two fixed points. The frequency of vibration is modified by changing the string length using a finger. The different strings have different wave speeds. When a string is plucked, a standing wave forms between the bridge and the finger.

                                                       

The string is displaced 0.4 cm at point P to sound the guitar. Point P on the string vibrates with simple harmonic motion (shm) in its first harmonic with a frequency of 195 Hz. The sounding length of the string is 62 cm.

Outline how a standing wave is produced on the string.

[2]
a.

Show that the speed of the wave on the string is about 240 m s−1.

[2]
b.i.

Sketch a graph to show how the acceleration of point P varies with its displacement from the rest position.

                 

[1]
b.ii.

Calculate, in m s−1, the maximum velocity of vibration of point P when it is vibrating with a frequency of 195 Hz.

[2]
b.iii.

Calculate, in terms of g, the maximum acceleration of P.

[2]
b.iv.

Estimate the displacement needed to double the energy of the string.

[2]
b.v.

The string is made to vibrate in its third harmonic. State the distance between consecutive nodes. 

[1]
c.

Markscheme

«travelling» wave moves along the length of the string and reflects «at fixed end» 

superposition/interference of incident and reflected waves

the superposition of the reflections is reinforced only for certain wavelengths  

a.

λ=2l=2×0.62=«1.24 m» ✓

v=fλ=195×1.24=242 «m s-1»

Answer must be to 3 or more sf or working shown for MP2.

b.i.

straight line through origin with negative gradient 

b.ii.

max velocity occurs at x = 0 

v=«(2π)(195)0.0042»=4.9 «m s1» 

b.iii.

a=2π 1952×0.004=6005 «m s2»

=600g 

b.iv.

use of EA2 OR  xo2 

A=0.42=0.57 «cm»  0.6 «cm» 

b.v.

623=21 «cm» 

c.

Examiners report

[N/A]
a.
[N/A]
b.i.
[N/A]
b.ii.
[N/A]
b.iii.
[N/A]
b.iv.
[N/A]
b.v.
[N/A]
c.

Syllabus sections

Additional higher level (AHL) » Topic 9: Wave phenomena » 9.1 – Simple harmonic motion
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Additional higher level (AHL) » Topic 9: Wave phenomena
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