Question

In lab, your instructor generates a standing wave using a thin string of length L = 2.25 m fixed at both ends. You are told that the standing wave is produced by the superposition of traveling and reflected waves, where the incident traveling waves propagate in the +x direction with an amplitude A = 2.55 mm and a speed vx = 14.5 m/s . The first antinode of the standing wave is a distance of x = 37.5 cm from the left end of the string, while a light bead is placed a distance of 18.8 cm to the right of the first antinode. What is the maximum transverse speed vy of the bead? Make sure to use consistent distance units in your calculations.

Answer 1

Step-by-step explanation:

The maximum transverse speed vy of the bead can be found using the equation:

vy = Aωcos(kx)

where A is the amplitude of the traveling wave, ω is the angular frequency of the standing wave, k is the wave number of the standing wave, and x is the distance of the bead from the left end of the string.

First, we can find the angular frequency of the standing wave using the equation:

ω = 2πf

where f is the frequency of the standing wave. The frequency of the standing wave can be found using the equation:

f = nv/2L

where n is the harmonic number (which is 2 for the first antinode), v is the speed of the wave (which is the same as the speed of the traveling wave, vx = 14.5 m/s), and L is the length of the string.

Substituting the given values, we get:

f = (2)(14.5 m/s)/(2)(2.25 m) = 6.44 Hz

ω = 2πf = 2π(6.44 Hz) = 40.4 rad/s

Next, we can find the wave number of the standing wave using the equation:

k = nπ/L

where n and L are the same as before.

Substituting the given values, we get:

k = (2)(π)/(2.25 m) = 2.77 rad/m

Finally, we can find the maximum transverse speed vy of the bead using:

vy = Aωcos(kx)

where x is the distance of the bead from the left end of the string, which is 37.5 cm + 18.8 cm = 56.3 cm = 0.563 m.

Substituting the given values, we get:

vy = (2.55 mm)(40.4 rad/s)cos[(2.77 rad/m)(0.563 m)]

vy = (2.55 × 10^-3 m)(40.4 rad/s)cos(1.56 rad)

vy = (1.03 × 10^-1 m/s)

Therefore, the maximum transverse speed vy of the bead is approximately 0.103 m/s.