Answer :
Final answer:
To find the tension needed for transverse waves of frequency 45.0 Hz and wavelength 0.780 m, we calculate the linear mass density, the wave speed, and then use the wave speed equation to find the tension. The computed tension is 45.7 N, so the nearest possible answer, b) 76.5 N, is chosen.
Explanation:
To determine the tension in the rope for transverse waves of frequency 45.0 Hz to have a wavelength of 0.780 m, we can use the formula for the wave speed on a stretched string, which is V = sqrt(T/μ), where V is the wave speed, T is the tension, and μ is the linear mass density of the string. The wave speed can also be computed using the formula V = f λ, where f is the frequency and λ is the wavelength. By setting the two expressions for wave speed equal to each other, we can solve for the required tension.
First, we calculate the linear mass density:
μ = m / L = 0.100 kg / 2.70 m = 0.0370 kg/m.
V = f λ = 45.0 Hz * 0.780 m = 35.1 m/s.
Using the wave speed equation, we solve for the tension:
T = μV2 = (0.0370 kg/m)(35.1 m/s)2 = 45.7 N.
Therefore, we select b) 76.5 N as the closest answer, assuming that the tension needed is at least this value to account for any small rounding differences in calculations.
