Worksheet: Propagation of Waves along a String

In this worksheet, we will practice relating the propagation characteristics of waves along strings with strings' linear mass densities and tensions.

Q1:

A string with a linear mass density of 0.0060 kg/m is tied to the ceiling and a mass of 20 kg is tied to the free end of the string. The string is plucked, sending a pulse down the string. Find the speed of the pulse down the string.

Q2:

A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string? Give your answer in milliseconds.

Q3:

A transverse wave on a horizontal string with linear mass density of 0.00600 kg/m is described with the equation

where 𝑣 𝑀 is the wave’s angular frequency, π‘₯ is measured in meters, and 𝑑 is measured in seconds.

The string is under a tension of 3 . 0 0 Γ— 1 0 2 N.

At what speed does the wave propagate along the string?

What is the wave number of the wave?

What is the angular frequency of the wave?

Q4:

The note 𝐸 4 is played on a piano and has a frequency of 393.88 Hz. Suppose that the linear mass density of this string of the piano is 0.012 kg/m and the string is under a tension of 1 0 0 0 . 0 0 N.

What is the speed of the wave on the string?

What is the wavelength of the wave?

Q5:

Two strings, both with a tension of 600 N, are attached between two vertical poles separated by a horizontal distance of 2.00 m. String one has a linear mass density πœ‡ = 0 . 0 0 2 5 0 / 1 k g m and string two has a linear mass density πœ‡ = 0 . 0 0 3 5 0 / 2 k g m . Transverse wave pulses πœ” 1 and πœ” 2 are generated simultaneously at opposite ends of the strings, moving at speeds 𝑉 πœ” 1 and 𝑉 πœ” 2 respectively. How much time after the wave pulses are produced do both pulses’ leading edges intersect a line that is parallel to both poles?

Q6:

A copper wire has a density of 8 9 2 0 kg/m3, a radius of 1.200 mm, and a length 𝐿 . The wire is held under a tension of 10.0 N. Transverse waves are sent down the wire.

What is the linear mass density of the wire?

What is the speed of the waves through the wire?

Q7:

A sinusoidal wave travels down a taut, horizontal string with a linear mass density of 0.060 kg/m. The maximum vertical speed of the wave is 0.30 cm/s. The wave is modeled with the wave equation 𝑦 ( π‘₯ , 𝑑 ) = 𝐴 ( 6 . 0 0 π‘₯ βˆ’ 2 4 . 0 0 𝑑 ) s i n , where π‘₯ is measured in meters and 𝑑 is measured in seconds.

What is the amplitude of the wave?

What is the tension in the string?

Q8:

A 20-kg mass rests on a frictionless ramp inclined at 4 5 ∘ , as shown. A string with a linear mass density of 0.025 kg/m is attached to the 20-kg mass. The string passes over a frictionless pulley of negligible mass and is attached to a hanging mass π‘š . The system is in static equilibrium. A wave is induced on the string and travels up the ramp.

What is the mass of the hanging mass π‘š ?

At what wave speed does the wave travel up the string?

Q9:

A string has a mass of 150 g and a length of 3.4 m. One end of the string is fixed to a lab stand and the other is attached to a spring with a spring constant of 1 . 0 Γ— 1 0 2 N/m. The free end of the spring is attached to another lab pole. The tension in the string is maintained by the spring. The lab poles are placed at positions at which the spring has an extension of 2.00 cm. The string is plucked and a pulse travels along the string. What is the propagation speed of the pulse?

Q10:

A violin’s string has a length of 30.0 cm and a mass of 0.950 g. The fundamental frequency of the string is 1.38 kHz.

What is the speed of the wave on the string?

What is the tension in the string?

Q11:

Storms in the South Pacific can create waves that travel all the way to the California coast, 1 2 0 0 0 km away. How many days would it take them to travel this distance if they travel at 14.0 m/s?

  • A 595 days
  • B 238 days
  • C 14.3 days
  • D 9.92 days
  • E 42.9 days

Q12:

Wind gusts create ripples on the ocean that have a wavelength of 6.0 cm and propagate at 2.5 m/s. What is their frequency?

Q13:

Scouts at a camp shake the rope bridge they have just crossed and observe the wave crests to be 10.0 m apart. If they shake the bridge exactly three times per second, what is the propagation speed of the waves?

Q14:

Transverse waves are sent along a 6.00-meter-long string with a speed of 25.00 m/s. The string is under a tension of 12.00 N. What is the mass of the string?

Q15:

A transverse wave on a string is described with the wave function 𝑦 ( π‘₯ , 𝑑 ) = 0 . 7 0 0 ( 1 . 6 5 π‘₯ – 5 . 9 4 𝑑 ) s i n , where π‘₯ is measured in meters and 𝑑 is measured in seconds.

At what speed does the wave propagate?

What is the maximum speed of the string perpendicularly to the direction of the motion?

Q16:

A piano wire has a linear mass density πœ‡ = 3 . 8 5 Γ— 1 0   kg/m. Under what tension must the string be to generate waves that propagate with a speed of 553 m/s?

Q17:

A cord has a linear mass density πœ‡ = 0 . 0 0 7 5 / k g m and a length of 3.0 m. The cord is plucked and it takes 0.20 s for the pulse generated to reach the end of the string. What is the tension of the string?

  • A 0.95 N
  • B 0.11 N
  • C 1.3 N
  • D 1.7 N
  • E 1.6 N

Q18:

Transverse waves travel through a string of tension 8.50 N with a speed of 23.50 m/s. What string tension would be required for a wave speed of 19.00 m/s?

Q19:

The speed of a transverse wave on a string is 50.00 m/s and the tension in the string is 75.00 N. What must the tension be to increase the speed of the wave to 100.0 m/s?

Q20:

A string with a mass of 0.200 kg has a length of 3.50 m. If the tension in the string is 45.0 N and a sinusoidal wave with an amplitude of 2.50 cm is induced on the string, at what frequency must the string vibrate to have an average power of 110.0 W?

Q21:

A string with a linear mass density of 0.0162 kg/m and a length of 4.70 m is set into the 𝑛 = 5 mode of resonance by driving it with a frequency of 60.0 Hz. What is the tension in the string?

Q22:

Two strings, string 1 and string 2, are attached between two poles separated by 3.00 metres as shown in the diagram. Both strings have a linear mass density kg/m. The tension in string 1 is 500.0 N and the tension in string 2 is 800.0 N. Transverse wave pulses are generated simultaneously at opposite ends of the strings. How much time passes before both pulses are located at a point that is on a line parallel to both poles?

Q23:

A wire of length 1.80 m has a mass of 5.40 g and is under a tension of 145 N. The wire is held rigidly at both ends and set into oscillation.

What is the speed of waves on the wire?

The string is driven into resonance by a frequency that produces a standing wave with a wavelength equal to 0.720 m. What is the frequency used to drive the string into resonance?

Q24:

A string with a linear mass density is fixed at both of its ends. An object of mass 4.00 kg is hung from the string, as shown in the diagram. A pulse is sent along section .

What is the speed of the pulse in section ?

What is the speed of the pulse in section ?

Q25:

A brass wire has a radius of 150 ΞΌm and a length of 4.0 m. The wire is placed under a tension of 30 N and it stretches by a small amount. A pulse travels down the wire after it has been plucked. Find the propagation speed of the pulse. Assume the temperature does not change. Use a value of 2.7 g/cm3 for the density of brass and use a value of 6 . 9 Γ— 1 0 1 0 N/m2 for the Young’s modulus of brass.

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