Question

A transverse wave with an amplitude of 0.200 mm and a frequency of 420 Hz moves along a tightly stretched string with a speed of 1.96 x 104 cm/s. (a) If the wave can be modeled as y = A sin(kx – wt), what are A (in m), k (in rad/m), and a (in rad/s)? (b) What is the tension in the string, if u = 4.60 g/m? (Give your answer in N.)

Answer 1

The amplitude of the wave is 0.0002 m and the wave number is 0.0677 rad/m. The angular frequency is 2640π rad/s. The tension in the string is 17662 N.

Step-by-step explanation:

(a) In the given equation y = A sin(kx – wt), A represents the amplitude of the wave. Here, the amplitude is given as 0.200 mm. Converting this to meters, the amplitude is 0.0002 m.

k represents the wave number, which is related to the wavelength. The wave number can be calculated using the equation k = 2π/λ, where λ is the wavelength. Given the frequency of 420 Hz, the wavelength can be found using the equation v = fλ, where v is the speed of the wave. Rearranging the equation, λ = v/f. Now we can calculate the wave number: k = 2π/(v/f) = 2πf/v = 2π * 420 / (1.96 x 10^4 * 10^-2) = 0.0677 rad/m.

a represents the angular frequency, which is related to the frequency. The angular frequency can be calculated using the equation a = 2πf. Therefore, the angular frequency is 2π * 420 = 2640π rad/s.

(b) The tension in the string can be calculated using the equation FT = u * v^2, where FT is the tension, u is the linear mass density, and v is the speed of the wave. Rearranging the equation, FT = u * (v^2). Given the linear mass density u as 4.60 g/m, we need to convert it to kg/m (1 g = 0.001 kg). FT = 0.0046 kg/m * (1.96 x 10^4 cm/s)^2 = 17662 N.

Answer 2

The amplitude A is 2 x 10^-4 m, wave number k is 13.49 rad/m, and angular frequency ω is 2638.94 rad/s. To find the tension in the string, use the wave speed formula: T = μv^2, which gives a tension T of 176.656 N.

Step-by-step explanation:

To find the wave parameters given a transverse wave with an amplitude of 0.200 mm, a frequency of 420 Hz, and a speed of 1.96 x 104 cm/s, we first need to convert the provided units to standard SI units. The amplitude in meters is 0.200 mm = 2 x 10-4 m, and the wave speed in meters per second is 1.96 x 104 cm/s = 196 m/s.

The amplitude A is therefore 2 x 10-4 m. To calculate the wave number k, we use the relationship k = 2π/λ, where λ is the wavelength. The wavelength can be found using the wave speed v and frequency f as λ = v/f. Thus, k = 2π/(v/f) = (2πf)/v. Substituting the given values, k = (2π * 420 Hz) / 196 m/s = 13.49 rad/m. The angular frequency ω is given by ω = 2πf, and substituting f = 420 Hz gives ω = 2π * 420 Hz = 2638.94 rad/s.

To determine the tension in the string, we can use the formula for the speed of a wave on a string, v = √(T/μ), where T is the tension and μ is the linear mass density. Rearranging for T gives T = μv2. Substituting in the values, T = (4.60 x 10-3 kg/m)(196 m/s)2 = 176.656 N.