Section 7: Measurements
1. Propagation of Error I
The radius of a sphere is measured to be \(r = (6.20 \pm 0.05)\text{ cm}\). Calculate the volume of the sphere and its associated uncertainty.
2. Propagation of Error II
The length and width of a rectangular plate are measured to be \(L = (15.3 \pm 0.1)\text{ cm}\) and \(W = (8.4 \pm 0.1)\text{ cm}\). Calculate the area of the plate and its uncertainty.
3. Propagation of Error III
The resistance R is calculated using Ohm's Law, \(R = V/I\). If the voltage is measured as \(V = (10.0 \pm 0.2)\text{ V}\) and the current as \(I = (2.00 \pm 0.05)\text{ A}\), what is the calculated resistance and its uncertainty?
4. Relative Uncertainty
A car's speedometer has a 5% of uncertainty. If it reads 60 km/h, what is the range of the car's actual speed?
5. Percentage Calculation
A measurement of time is recorded as \(t = 5.45 \pm 0.22\) seconds. What is the percentage uncertainty of this measurement?
6. Instrument Precision
A digital thermometer reads \(25.4^\circ\text{C}\). Assuming the uncertainty is half the value of the last digit, what is the absolute uncertainty of this measurement?
7. Standard Deviation
Eleven students received the following scores on a test: 88, 92, 79, 85, 95, 81, 86, 90, 83, 77, 89. What is the mean $\bar{x}=\frac{1}{N} \sum_{i=1}^N x_i $ and standard deviation $\sigma=\sqrt{\frac{1}{N-1} \sum_{i=1}^N (x_i - \bar{x})^2} $ of these test scores? If the highest and lowest scores are removed, what are the new mean and standard deviation of the remaining scores?
8. Mass-Spring Measurements - html
Generate a simulator of a mass suspended on a spring in HTML with a timing function. Treating the mass as a given value with zero uncertainty, perform a series of 10 time measurements for 10 complete oscillations. Use the collected data to calculate the mean period, standard deviation. Calculate the value of the spring constant along with its measurement uncertainty.
9. Pendulum Measurements - html/real
Create a simple pendulum simulator in HTML equipped with a stopwatch for manual timing. Assuming the string length is an exact value, run the simulation and perform 10 measurements of the time taken for 10 complete oscillations. Based on the obtained results, manually calculate the mean period and standard deviation. Using this data, determine the value of the acceleration due to gravity and calculate the measurement uncertainty of this result.
Optionally, repeat the real-life experiment: set up a physical pendulum using a string and a small mass (e.g., a metal ball, keychain, necklace). Measure the length of the pendulum and perform 10 measurements of the time taken for 10 complete oscillations using a stopwatch in your cellphone. Calculate the mean period and its standard deviation. Use this data to calculate the value of the acceleration due to gravity and its associated uncertainty.
10. Light Speed Measurement
Measure the speed of light using a microwave oven, a bar of chocolate (or slices of cheese), and the ruler. To this end, measure the distance between the melted spots on the chocolate to determine the wavelength of the microwaves. Use the frequency of the typical microwave oven, f=2.45 GHz, to calculate the speed of light. How does your result compare to the accepted value of c = 300 000 000 m/s? What is the percentage error of your measurement?