Question 2(c) Merit to Excellence

On a hot day the violin goes out of tune easily, and Sam has to adjust the tension in the "G" string to keep it vibrating at 196 Hz.

Sam hears a beat of 2.1 Hz. She then increases the speed of the wave along the string by tightening it, and the beat frequency increases.

(i) Determine the possible frequencies of the string. (ii) Use the tightening to decide which frequency it was vibrating at before adjustment. (iii) Explain what Sam must do to get the string to 196 Hz. (iv) State how she will know when it is vibrating at 196 Hz.

The 2023 paper, page 4 ↗ Marking schedule, page 3 ↗

Work it out

Try it on paper first. Stuck? Take one hint at a time.

1. 1 (i) A beat of 2.1 Hz is the difference between the two frequencies. What could the string be at?

   f_beat = |f − 196| = 2.1, so the string is at 196 + 2.1 = 198.1 Hz or 196 − 2.1 = 193.9 Hz.

2. 2 (ii) Tightening raises the wave speed. What does that do to the string’s frequency?

   Raises it: the wavelength is fixed by the string’s length, so f = v / λ goes up.

3. 3 The frequency went up and the beats got faster. Which starting value fits?

   Faster beats mean the string moved further from 196 Hz. Going up and getting further away means it started above 196 Hz: 198.1 Hz. (Starting from 193.9 Hz, going up would have slowed the beats.)

4. 4 (iii) So what must Sam do?

   Bring the frequency down: loosen the string to reduce the tension, which lowers the wave speed and so the frequency, until it is 196 Hz.

5. 5 (iv) How will she know?

   The beats slow down as she loosens it, and when the string is exactly at 196 Hz they stop: she hears one steady note.

What the marker wanted

• Achieved Both possible frequencies, 198.1 and 193.9 Hz; or that there is no beat when the string is exactly at 196 Hz.
• Merit Justifies the rise in frequency in (ii), or the lower tension in (iii).
• Excellence The complete answer, (i) to (iv).

The schedule marks (c)(i) to (iv) together as one item.

In our words, from NZQA’s 2023 marking schedule.

How did it go?

Got it moves it further out. Review brings it back tomorrow.