**Unit 4: Sound and Music**

**Problem Set B**

The following selection of problems are sample problems. Individual student problem sets will vary since numerical information is randomly-generated.

**Directions:**

For the following problems:

- Compute the unknown quantity and enter the answer in the blank.
- Give regard to positive and negative values and include the sign in the answer (if negative).
- Do not round any computed numbers until the last calculation.
- Enter your answers accurate to the second decimal place (unless told otherwise).

**Problem 1: **

A circus performer stretches a tightrope between two towers. He strikes one end of the rope and sends a wave along it toward the other tower. He notes that it takes the wave 0.685 s to reach the opposite tower 20 m away. If one meter of the wire has a mass of 0.37 kg, find the tension (in Newtons) in the tightrope.

**Problem 2:**

A phone cord is 2.06 m long. The cord has a mass of 0.177 kg. If a transverse wave pulse travels from the receiver to the phone box in 0.112 s, what is the tension (in Newtons) in the cord?

**Problem 3: **

Transverse waves with a speed of 47.3 m/s are to be produced on a stretched string. A 4.94-m length of string with a total mass of 0.058 kg is used. What is the required tension (in Newtons) in the string?

**Problem 4: **

(Referring to the previous problem.) Calculate the wave speed (in m/s) in the string (assuming the same mass and length) if the tension is 7.64 N.

**Problem 5: **

One end of a 3.01-meter long string is attached to a wall while the other end hangs over a pulley and is attached to a hanging 2.0-kg mass (which creates a tension of 19.6 N). The speed of the pulse on the string is observed to be 14.8 m/s. What is the mass (in grams) of the string?

**Problem 6: **

A physics teacher stretches a string to a length of 1.04 meters and uses a mechanical oscillator to vibrate the string in its 6th harmonic. If the oscillator's frequency is 613 Hertz, then what is the speed (in meters/second) at which the waves travel through the string?

**Problem 7: **

(Referring to the previous problem.) The same string held at the same tension is vibrated at a frequency of 306.5 Hertz to create a different standing wave pattern. Determine the wavelength (in centimeters) of waves for this particular frequency.

**Problem 8: **

A steel wire in a piano has a length of 0.683 m and a mass of 4.24
X 10^{-3} kg. To what tension (in Newtons) must this wire be
stretched in order that the fundamental vibration correspond to
middle C (frequency of middle C = 261.6 Hz on the chromatic musical
scale)?

**Problem 9: **

A stretched string is 153 cm long and has a linear density of 0.014 g/cm. What tension (in Newtons) in the string will result in a second harmonic of 455 Hz?

**Problem 10: **

A wire of mass .276 g is stretched between two points 69.7 cm apart. If the tension in the wire is 563 N, find the frequency (in Hertz) of the 5th harmonic.

**Problem 11: **

(Referring to the previous problem.) Determine the frequency (in Hertz) of the fundamental for this wire.

**Problem 12: **

A stretched string fixed at each end has a mass of 38.7 g and a length of 7.24 m. The tension in the string is 55.7 N. What is the vibration frequency (in Hertz) for this third harmonic?

**Problem 13: **

Two pieces of steel wire having identical cross sections have lengths of L and 2L. The wires are each fixed at both ends and stretched such that the tension in the longer wire is four times greater than that in the shorter wire. If the fundamental frequency in the shorter wire is 63 Hz, what is the frequency (in Hertz) of the second harmonic in the longer wire? (HINT: first determine how many times greater the speed of the tighter wire is than the less tight wire.)

**Problem 14: **

A stretched string of length L is observed to vibrate in 6 equal segments (i.e., with 6 antinodes between its ends) when driven by a 605-Hz oscillator. What oscillator frequency (in Hertz) will set up a standing wave pattern such that the string vibrates in 3 segments (3 antinodes)?

**Problem 15: **

A standing wave pattern is established in a string that is 234 cm long and fixed at both ends. The string vibrates in 4 segments (i.e., with 4 antinodes between its ends) when driven at 147 Hz. Determine the wavelength (in centimeters).

**Problem 16: **

(Referring to the previous problem.) What is the fundamental frequency (in Hertz) of this string?

**Problem 17: **

A string 51.2 cm long has a mass per unit length equal to 14 X
10^{-5} kg/m. To what tension (in Newtons) should this string
be stretched if its fundamental frequency is to be 35 Hz? Enter your
answer accurate to the fourth decimal place.

**Problem 18: **

(Referring to the previous problem.) To what tension (in Newtons) should this string be stretched if its fundamental frequency is to be 3090 Hz?

**Problem 19: **

The fundamental frequency of a 161-cm section of string is 145 Hertz. Determine the frequency (in Hertz) of the 6th harmonic of a 80.5-cm section of the same string. (Hint: first calculate the speed of waves in this string.)

**Problem 20: **

When held at a tension of **T**, a 111-cm long string has a
fundamental frequency of 239 Hertz. Suppose that the tension is
increased by a factor of 2.9 without any alteration in the string
length or mass per unit length. What would be the new fundamental
frequency (in Hertz)?

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