A student investigates how the speed of transverse waves on a single uniform string depends on the tension applied to the string. The student hangs different masses from one end of the string that passes over a pulley, producing the tensions shown in the table. For each tension the student measures the wave speed by creating a pulse and timing its travel over a fixed distance.
Trial
Hanging mass (kg)
Tension FT (N)
Measured speed v (m/s)
1
1.0
9.8
20
2
2.0
19.6
28.3
3
4.0
39.2
40
Assume the linear mass density of the string remained constant during the trials.
(a) Using the numerical data, determine whether the wave speed is proportional to the square root of the applied tension. Show your reasoning explicitly.
(b) Predict the wave speed for a tension of FT=78N and explain the reasoning or calculation that leads to your prediction.
(c) Suppose the experiment were repeated with a second string whose linear density is exactly half that of the original string, while all hanging masses remain the same. Qualitatively describe how the measured speeds in each trial would compare with those listed above, and justify your answer without doing explicit numerical calculations.