## Questions

1) Calculate the volume of the bottom of the cylindrical can in Fig. 0.1. The outside measures cm, the inside measures cm, and the diameter measures cm. Assume the errors are independent. Write down your result and its error, using:

a) Absolute error.
b) Fractional error (as a percentage).
c) Significant figures.

FIGURE 0.1

2) Students in T&AM 203 (Dynamics) can choose between two different procedures when measuring the period of an oscillating weight on a spring. In Procedure A, a timer is started and stopped electronically, as the weight repeatedly passes through a light beam; the error in starting and stopping the timer is negligible. The timer reads in units of 1 millisecond, but is not crystal-controlled, and typically runs 2 percent fast or slow.

In Procedure B a regular hand-held digital stopwatch is used, which reads in units of 0.01 seconds, or 10 ms. Being crystal-controlled, the stopwatch runs at the correct speed, and is not significantly fast or slow. However, since it is manually operated, an error of about 0.1 second is introduced once as the stopwatch is started and again when it is stopped (assume these errors are independent).

Assume that the period of oscillation of the weight is exactly two seconds.

a) If Procedures A and B are each used to measure one oscillation of the weight, what are their respective precisions? Their accuracies?
b) If Procedure A is used to measure 10 oscillations of the weight, and the resulting ten measured periods are averaged, what are the precision and accuracy of the average?
c) If Procedure B is used to measure 10 oscillations of the weight, and the resulting ten measured periods are averaged, what are the precision and accuracy of the average?
d) Suppose that the stopwatch from Procedure B is used, but somewhat differently: the stopwatch is started, ten oscillations of the weight are counted, and then the stopwatch is stopped. The resulting time measurement is divided by ten. Find
the accuracy and precision, and compare them to the previous values.
e) Which procedure has the largest systematic error?

3) A carpenter laying out a 40 foot wall with studs every 16 inches can't find his long measuring tape, so he carefully cuts a scrap board to a length of 16 0.05 inches, and starting from one end, repeatedly lays it down along the floor to mark out the locations for the studs. Assuming that there are no other sources of error in the marking process, estimate the error in the position of the final stud.

4) Ultrasonic range finders bounce high-frequency pulses of sound off objects, and time the echoes to determine distance. One popular model has a range of 0.3 to 10 meters, and (at an air temperature of 20ºC) gives an echo time of 0.0017 seconds at a distance of 0.3 meters, and 0.0583 seconds at 10 meters. Based on the way it works, do you think the device is linear? (Assume uniform air properties.) If possible, calibrate the range finder by finding a formula relating echo time to distance. Optional: What else could you measure with this device?

 Introduction Main Body •Experimental Error •Minimizing Systematic Error •Minimizing Random Error •Propagation of Error •Significant Figures Questions