Q: A student measures the density of a metal and finds $$\rho = 7.53 \pm 0.05\ \mathrm{g/cm^3$}$$, while the accepted value is $$7.87 \pm 0.01\ \mathrm{g/cm^3$}. $$What is the percentage discrepancy between the measured and accepted values?

$$\displaystyle \frac{7.53 - 7.87}{7.87} \times 100\% = -4.3\%$$

Q: A measurement is reported as $$L = 2.50 \pm 0.05 m. $$What is the percentage uncertainty?

$$\displaystyle \frac{0.05}{2.50} \times 100\% = 2.0\%$$

Question 1

Which of the following best describes the purpose of plotting measured values with their uncertainties on a comparison graph, as shown in the images above?

Accepted Answer

To visually determine if the ranges of measured values overlap, indicating experimental agreement.

Question 2

A student measures the density of a metal and finds $$\rho = 7.53 \pm 0.05\ \mathrm{g/cm^3$}$$, while the accepted value is $$7.87 \pm 0.01\ \mathrm{g/cm^3$}. $$What is the percentage discrepancy between the measured and accepted values?

Accepted Answer

$$\displaystyle \frac{7.53 - 7.87}{7.87} 	imes 100\% = -4.3\%$$

Question 3

When combining uncertainties for a calculated density $$\rho = \frac{m}{LWH}$$, which method should be used?

Accepted Answer

Add the percentage uncertainties of $$m$$, $$L$$, $$W$$, and $$H in $$quadrature: $$\%\rho = \sqrt{(\%m)^2$ + (\%L)^2$ + (\%W)^2$ + (\%H)^2$}$$

Question 4

A block is measured with a ruler (precision $$\pm 0.1 cm$$, accuracy $$\pm 0.05 cm). $$What is the total uncertainty in a single length measurement?

Accepted Answer

$$\Delta \ell = \sqrt{(0.1)^2$ + (0.05)^2$} = 0.11 cm$$

Question 5

Which of the following statements about significant figures is correct?

Accepted Answer

The first doubtful digit is the last digit to be retained in a reported value.

Question 6

A measurement is reported as $$L = 2.50 \pm 0.05 m. $$What is the percentage uncertainty?

Accepted Answer

$$\displaystyle \frac{0.05}{2.50} 	imes 100\% = 2.0\%$$

Question 7

Which of the following best describes the difference between accuracy and precision?

Accepted Answer

Accuracy is how close a measurement is to the true value; precision is how repeatable the measurement is.

Question 8

If a block's length is measured as $$L = 10.00 \pm 0.05 cm$$, width as $$W = 5.00 \pm 0.02 cm$$, and height as $$H = 2.00 \pm 0.01 cm$$, what is the percentage uncertainty in the calculated volume $$V = LWH$$?

Accepted Answer

$$\%V = \sqrt{(0.5)^2$ + (0.4)^2$ + (0.5)^2$} = 0.85\% ($$where $$\%L = 0.5\%$$, $$\%W = 0.4\%$$, $$\%H = 0.5\%)$$

Question 9

Which of the following is the correct way to report a measurement and its uncertainty?

Accepted Answer

$$L = 2.50 \pm 0.05 m$$

Question 10

A student calculates the density of a cylinder using $$\rho = \frac{4m}{\pi d^2$ l}. If $$the percentage uncertainty in $$d is$$ $$0.5\%$$, what is the contribution of $$d to $$the percentage uncertainty in $$\rho$$?

Accepted Answer

$$4 	imes 0.5\% = 2.0\%$$

Question 11

Which of the following is NOT a valid reason for a discrepancy between a measured and accepted value in a density experiment?

Accepted Answer

The block was measured using the same instrument for all dimensions.

Question 12

When reporting a value in scientific notation, which of the following is correct for a value with four significant figures?

Accepted Answer

$$9.163 	imes 10^2$

Question 13

If a measurement is repeated several times and the results are 1.760 m, 1.775 m, 1.765 m, and 1.780 m, what is the best estimate of the uncertainty?

Accepted Answer

Half the range: $$(1.780 - 1.760)/2 = 0.01 m$$

Question 14

Which of the following is the correct formula for the slope of a straight line on a graph?

Accepted Answer

$$B = \frac{y_2$ - y_1$}{x_2$ - x_1$}$$

Question 15

In a graph of $$y$$ versus $$x$$, the best-fit line does not pass through all data points. What does this indicate?

Accepted Answer

The line shows the trend of the data, averaging out random errors.

Applied Physics Laboratory Foundations: Measurement, Uncertainty, Graphical Analysis, Newton’s Laws, Energy, and Thermal Physics

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