Final answer:
The new wave speed on the string after decreasing the tension by a quarter is approximately 34.6 m/s, calculated by taking the square root of 3/4 of the original tension over the linear mass density and multiplying by the original speed of 40.0 m/s.
Step-by-step explanation:
The speed of a wave on a string is related to the tension (T) in the string and the linear mass density (μ) of the string. The speed of the wave (v) is given by the equation v = √(T/μ). If the tension is decreased by a quarter, the new tension T' is T' = T - T/4 = 3T/4. Substituting this into our speed equation gives us the new speed v' = √(3T/4μ). To compare the new speed with the old speed, we can express this relationship as a ratio of the new speed to the old speed:
v'/v = √(3T/4μ) / √(T/μ) = √(3/4) = √0.75 ≈ 0.866. So, the new speed is approximately 86.6% of the original speed. If the original speed was 40.0 m/s, then the new speed is 0.866 × 40.0 m/s ≈ 34.6 m/s. Therefore, the new wave speed on the string when the tension is decreased by a quarter is not explicitly given in the options a) to d), but it is closest to 34.6 m/s.