Final answer:
The string density uncertainty is calculated by assessing the precision of the instruments used to measure mass and length, determining the absolute uncertainty, and then applying error propagation rules to the linear mass density calculation. The velocity of the wave on the string can be calculated using the equation v = √(T/μ), and the uncertainty in any derived quantity, like average speed, follows from the uncertainties in the measurements of mass, length, tension, and time.
Step-by-step explanation:
Calculating String Density Uncertainty
To estimate the speed of a pulse moving down a string, we use the formula for the velocity of a wave on a string v = √(T/μ), where T is the tension in the string and μ (mu) is the linear mass density of the string. For instance, if a 20-kg mass is attached to a string with a μ of 0.0060 kg/m, the tension T is equal to the weight of the mass (T = m*g = 20kg * 9.8m/s²). To calculate the uncertainty of the string density, we must consider the precision of the instruments used to measure mass and length. If the smallest divisions on the measurement devices are known, we can determine the uncertainty by using these values as the possible error in our measurements. The percent uncertainty in a quantity is calculated by dividing the absolute uncertainty by the measured value and multiplying by 100. This concept is also applied to average speed and any other measured value, such as the tension in the string or the elapsed time.
For example, if the mass of the string is measured to be 5.00 g (0.005 kg) with a maximum possible error of 0.01 g and the length is 3.00 m with a possible error of 0.001 m, then the linear mass density μ is 0.005 kg / 3.00 m = 0.00167 kg/m. The uncertainty in μ could be expressed by calculating the uncertainties in both the mass and length measurements and then applying error propagation rules to find the combined uncertainty in μ.