Translation between representations (TBR)

Standing waves on a string held taut by a hanging block

Unit 14
12 points
About 28 minutes

You can use a calculator on this question, just like on exam day.

A multipart problem about one situation shown in several ways. You draw a diagram, derive equations, sketch or draw graphs, and then explain whether your answers agree with each other or use them to predict what happens when the situation changes. On the exam: Question 2 of 4. New for May 2027: the free-response section is 95 minutes, down from 100, for all 4 questions (50% of your score), one of each type in this order. Calculator and equation sheet allowed. The CED suggests 25–30 minutes. This question type started in May 2025, so Physics 2 free-response questions from 2024 and earlier are built differently.

The question

One end of a horizontal string is tied to a mechanical vibrator whose frequency can be adjusted. The string runs horizontally for 1.20 m, passes over a light, frictionless pulley and is tied to a hanging block of mass M, which keeps the string taut. The vibrating section of string, from the vibrator to the top of the pulley, is 1.20 m long, and both of its ends act as nodes. The string's mass per unit length is 5.0 × 10⁻³ kg/m. Use g = 9.8 m/s².

Suggested time: 28 minutes

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Part (a)

3 points

With M = 2.0 kg, the vibrator is adjusted so that the string vibrates in its third harmonic. Describe the standing wave you would sketch: the number and positions of the nodes and antinodes, the wavelength, and how the motion of the string in neighboring loops compares.

Your answer to part (a)

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Part (b)

3 points

Derive an expression for the frequency $f_n$ of the nth harmonic of the string in terms of n, the vibrating length L, the hanging mass M, the mass per unit length μ, and physical constants, as appropriate.

Your answer to part (b)

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Part (c)

3 points

Describe the graph of the fundamental frequency $f_1$ of the string as a function of the hanging mass M, for M from 0 to 4.0 kg. Include the axes with units, the shape of the graph, and the values of $f_1$ at M = 2.0 kg and M = 4.0 kg.

Your answer to part (c)

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

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Part (d)

3 points

The vibrator is kept at the frequency that produced the third harmonic with M = 2.0 kg. A student slowly increases M until a new standing-wave pattern forms. (i) Will the new pattern have more loops or fewer loops? Justify your claim. (ii) Calculate the value of M for the new pattern. (iii) Explain how your answer is consistent with your graph from part (c).

Your answer to part (d)

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

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