Measurement uncertainty and error Transitional Reasoning Practice

Questions

1. 1.

   A student measures the uniform speed of a motorized cart on a linear track using an electronic timer that has a fixed absolute uncertainty of $\pm 0.04\text{ s}$, while the distance measurement has negligible uncertainty. In the first trial, the cart covers a short distance in $(0.80 \pm 0.04)\text{ s}$, whereas in the second trial, it covers a longer distance in $(4.00 \pm 0.04)\text{ s}$. Which choice correctly gives the percentage uncertainty in the calculated speed for each trial and states the experimental principle demonstrated by this comparison?

   • A) 2.0% and 0.4%; increasing the measurement duration decreases the timer's absolute uncertainty.
   • B) 5.0% and 5.0%; the percentage uncertainty remains constant because the same timing instrument is used in both trials.
   • C) 10.0% and 2.0%; measuring across a longer distance decreases the absolute uncertainty of the timing sensor.
   • D) 5.0% and 1.0%; measuring a larger value with a fixed absolute uncertainty reduces the percentage uncertainty.
2. 2.

   To investigate the impact speed of a drop test capsule, an engineering student releases the capsule from a measured vertical height of $h = (5.00 \pm 0.30)\text{ m}$. The impact speed is modeled by the theoretical relation $v = \sqrt{2gh}$, where the gravitational acceleration is taken as a constant $g = 10.0\text{ m/s}^2$ with negligible uncertainty. According to the principles of experimental uncertainty propagation, which choice correctly gives the percentage uncertainty in the calculated impact speed and reports the speed in standard $(v \pm \Delta v)$ form?

   • A) 6.0% and $(10.0 \pm 0.6)\text{ m/s}$
   • B) 3.0% and $(10.0 \pm 0.3)\text{ m/s}$
   • C) 12.0% and $(10.0 \pm 1.2)\text{ m/s}$
   • D) 1.5% and $(10.0 \pm 0.15)\text{ m/s}$
3. 3.

   To evaluate the performance of an electric lifting mechanism, a student measures the electrical energy supplied to the motor as $E_{\text{in}} = (200 \pm 6)\text{ J}$ and the gravitational potential energy gained by the lifted load as $E_{\text{out}} = (120 \pm 6)\text{ J}$. The efficiency of the system is defined by the relationship $\eta = \frac{E_{\text{out}}}{E_{\text{in}}}$. According to the rules of experimental uncertainty propagation for quotients, what are the percentage uncertainty in the calculated efficiency and the efficiency reported in the form $(\eta \pm \Delta \eta)$?

   • A) Percentage uncertainty: 2.0%; reported efficiency: $(60.0 \pm 1.2)\%$
   • B) Percentage uncertainty: 8.0%; reported efficiency: $(60.0 \pm 12.0)\%$
   • C) Percentage uncertainty: 5.0%; reported efficiency: $(60.0 \pm 3.0)\%$
   • D) Percentage uncertainty: 8.0%; reported efficiency: $(60.0 \pm 4.8)\%$
4. 4.

   To determine the spring constant $k$ of an elastic cord, a student measures the tensile force $F$ across multiple extensions $x$ and plots a graph of $F$ versus $x$. The line of best fit yields a gradient of 50.0 N/m, while the steepest and shallowest trendlines consistent with the error bars give gradients of 53.0 N/m and 47.0 N/m, respectively. Based on standard graphical uncertainty analysis, which choice correctly reports the spring constant in the form $(k \pm \Delta k)$ along with its percentage uncertainty?

   • A) $(50.0 \pm 3.0)\text{ N/m}$ and 6.0%
   • B) $(50.0 \pm 6.0)\text{ N/m}$ and 12.0%
   • C) $(50.0 \pm 3.0)\text{ N/m}$ and 3.0%
   • D) $(50.0 \pm 1.5)\text{ N/m}$ and 3.0%
5. 5.

   To determine the wall thickness of a cylindrical pipe, a student measures its outer diameter as $D_2 = (50.0 \pm 0.2)$ mm and inner diameter as $D_1 = (46.0 \pm 0.2)$ mm. The wall thickness is given by $t = \frac{1}{2}(D_2 - D_1)$. Based on the rules of uncertainty propagation, which choice correctly reports the wall thickness in standard form $(t \pm \Delta t)$ along with its percentage uncertainty?

   • A) $t = (2.0 \pm 0.2)$ mm with a percentage uncertainty of 10%
   • B) $t = (2.0 \pm 0.4)$ mm with a percentage uncertainty of 20%
   • C) $t = (4.0 \pm 0.4)$ mm with a percentage uncertainty of 10%
   • D) $t = (2.0 \pm 0.1)$ mm with a percentage uncertainty of 5.0%
6. 6.

   A student must determine the volume of a uniform solid sphere with an approximate diameter of 20.0 mm for an engineering project. The experimental design requires that the percentage uncertainty in the calculated volume does not exceed 3.0%. Based on the relationship $V = \frac{1}{6}\pi d^3$, what are the maximum allowable percentage uncertainty and the maximum allowable absolute uncertainty in the diameter measurement?

   • A) Maximum percentage uncertainty: 3.0%; maximum absolute uncertainty: $\pm 0.60\text{ mm}$
   • B) Maximum percentage uncertainty: 0.33%; maximum absolute uncertainty: $\pm 0.067\text{ mm}$
   • C) Maximum percentage uncertainty: 1.0%; maximum absolute uncertainty: $\pm 0.20\text{ mm}$
   • D) Maximum percentage uncertainty: 9.0%; maximum absolute uncertainty: $\pm 1.80\text{ mm}$
7. 7.

   A student uses an analog ammeter with scale divisions marked every 0.2 A to measure the current in a direct-current circuit. Across five repeated trials under identical conditions, the pointer rests directly on the 3.6 A mark each time with no visible fluctuation. Based on this observation, the student concludes that the measurement has zero uncertainty and records the result as $3.60\text{ A} \pm 0.00\text{ A}$. Which statement correctly evaluates the student's conclusion and specifies how the result should be reported?

   • A) The conclusion is correct because identical readings across multiple trials confirm that random fluctuations and reading errors are zero, validating the report of $3.60\text{ A} \pm 0.00\text{ A}$.
   • B) The conclusion is incorrect because identical readings only indicate that random variations are smaller than the instrument's resolution; the uncertainty is limited by the scale reading uncertainty of half the smallest division, giving $(3.6 \pm 0.1)\text{ A}$.
   • C) The conclusion is incorrect because repeating identical values proves the presence of a systematic zero offset equal to one full division, requiring the result to be corrected to $(3.8 \pm 0.2)\text{ A}$.
   • D) The conclusion is incorrect because repeated trials on an analog scale require dividing the smallest division by the number of trials, yielding a reported current of $(3.60 \pm 0.04)\text{ A}$.
8. 8.

   A student investigates the resistance $R$ of an ohmic resistor by measuring the potential difference $V$ across it at several distinct currents $I$. Unbeknownst to the student, the voltmeter has a constant zero error that adds $+0.4\text{ V}$ to every reading across all trials. The student considers two methods to determine $R$: averaging the individual ratios $\frac{V}{I}$ across trials, or finding the gradient of a best-fit line on a graph of $V$ versus $I$. Which statement correctly evaluates how this systematic error affects the calculated resistance in each method?

   • A) Both methods yield an identical overestimate of $R$ because a constant systematic error propagates equally into algebraic averages and graphical slopes.
   • B) The gradient method yields an accurate value of $R$ because the constant offset alters only the vertical intercept, whereas averaging the ratios overestimates $R$.
   • C) Averaging the ratios yields an accurate value of $R$ because repeating trials cancels the offset, whereas the gradient method systematically overestimates $R$.
   • D) Both methods yield an accurate value of $R$ because collecting data across a range of currents neutralizes the constant voltage offset.
9. 9.

   Two laboratory groups, Group X and Group Y, each perform five repeated trials to determine the boiling point of a pure liquid sample with an accepted reference value of 78.0°C. Group X records readings of 81.9°C, 82.1°C, 82.0°C, 82.0°C, and 82.0°C, while Group Y records readings of 76.8°C, 79.1°C, 77.5°C, 79.4°C, and 77.2°C. Based on the principles of experimental error and measurement uncertainty, which statement correctly evaluates the accuracy and precision of both groups and identifies the predominant error type in Group X's data?

   • A) Group X has higher accuracy than Group Y because its measurements are more consistent, and its offset is caused by random error.
   • B) Group X has both higher accuracy and higher precision than Group Y because all its readings are clustered within a 0.2°C range.
   • C) Group X has higher precision but lower accuracy than Group Y, and the offset in Group X's mean is caused by a systematic error.
   • D) Group Y has higher precision but lower accuracy than Group X, and increasing the number of trials will eliminate Group X's offset.
10. 10.

    A glider on a linear air track is released from rest and undergoes uniform acceleration. A sensor system records its displacement as $s = (1.80 \pm 0.09)\text{ m}$ and the time taken as $t = (3.00 \pm 0.15)\text{ s}$. Using the kinematic relationship $a = \frac{2s}{t^2}$, which of the following correctly reports the calculated acceleration in standard form $(a \pm \Delta a)$ along with its percentage uncertainty?

    • A) $a = (0.40 \pm 0.02)\text{ m/s}^2$ with a percentage uncertainty of $5\%$
    • B) $a = (0.40 \pm 0.04)\text{ m/s}^2$ with a percentage uncertainty of $10\%$
    • C) $a = (0.40 \pm 0.08)\text{ m/s}^2$ with a percentage uncertainty of $20\%$
    • D) $a = (0.40 \pm 0.06)\text{ m/s}^2$ with a percentage uncertainty of $15\%$
11. 11.

    A physics student determines the density of a standardized metal alloy sample as $(7.60 \pm 0.19)\text{ g/cm}^3$, while the manufacturer's certified reference value is $8.00\text{ g/cm}^3$. To evaluate the reliability of the result, the student calculates both the percentage discrepancy relative to the reference value and the experimental percentage uncertainty of the measurement. What are the percentage discrepancy and the percentage uncertainty, and what conclusion should be drawn regarding the experimental error?

    • A) Percentage discrepancy: 2.5%; percentage uncertainty: 5.0%; the discrepancy is fully accounted for by random errors alone.
    • B) Percentage discrepancy: 5.0%; percentage uncertainty: 2.5%; the discrepancy is fully accounted for by random errors because the measured value is close to the reference value.
    • C) Percentage discrepancy: 4.0%; percentage uncertainty: 1.9%; systematic errors are negligible because the absolute difference is less than $1.0\text{ g/cm}^3$.
    • D) Percentage discrepancy: 5.0%; percentage uncertainty: 2.5%; the discrepancy cannot be explained by random uncertainty alone, indicating the presence of a systematic error.
12. 12.

    A student conducts an experiment to test whether momentum is conserved during a collision on a low-friction track. The measured total initial momentum is $p_{\text{initial}} = (2.40 \pm 0.08)\text{ kg}\cdot\text{m/s}$, and the measured total final momentum is $p_{\text{final}} = (2.28 \pm 0.06)\text{ kg}\cdot\text{m/s}$. Based on experimental uncertainty analysis, which statement correctly evaluates whether these measurements support the conservation of momentum?

    • A) The measurements do not support conservation because the central value of the final momentum is lower than that of the initial momentum.
    • B) The measurements support conservation because the difference of $0.12\text{ kg}\cdot\text{m/s}$ between the central values is less than the combined absolute uncertainty of $0.14\text{ kg}\cdot\text{m/s}$.
    • C) The measurements support conservation because the percentage uncertainty of the final momentum is smaller than that of the initial momentum.
    • D) The measurements do not support conservation because the uncertainty intervals do not share any overlapping values.
13. 13.

    In a physics laboratory, a student measures the rectangular base of an aluminum heat sink using a vernier caliper and records the length as $L = (20.0 \pm 0.4)\text{ cm}$ and the width as $W = (10.0 \pm 0.3)\text{ cm}$. According to the rules of experimental uncertainty propagation for independent measurements, which of the following correctly expresses the area of the base in standard $(A \pm \Delta A)$ form?

    • A) $(200 \pm 0.12)\text{ cm}^2$
    • B) $(200 \pm 0.7)\text{ cm}^2$
    • C) $(200 \pm 5)\text{ cm}^2$
    • D) $(200 \pm 10)\text{ cm}^2$
14. 14.

    To determine the density of an irregularly shaped mineral sample, a student measures its mass as $(80.0 \pm 1.6)\text{ g}$. The volume of the sample is measured by water displacement in a graduated cylinder, where the water level rises from an initial reading of $(30.0 \pm 0.5)\text{ cm}^3$ to a final reading of $(50.0 \pm 0.5)\text{ cm}^3$. Based on these measurements, what are the percentage uncertainty in the sample's volume and the total percentage uncertainty in its calculated density?

    • A) Volume uncertainty: 2.5%, Density uncertainty: 4.5%
    • B) Volume uncertainty: 5.0%, Density uncertainty: 7.0%
    • C) Volume uncertainty: 5.0%, Density uncertainty: 3.0%
    • D) Volume uncertainty: 1.0%, Density uncertainty: 2.6%
15. 15.

    A student uses a stopwatch with an estimated reaction-time uncertainty of $\pm 0.40\text{ s}$ to determine the period of a simple pendulum. To reduce the uncertainty associated with a single cycle, the student measures the total duration of $20$ consecutive complete oscillations and records a value of $(38.00 \pm 0.40)\text{ s}$. Based on these experimental data, what are the period of one oscillation and its associated absolute uncertainty?

    • A) $T = 1.90\text{ s}$ with $\Delta T = \pm 0.40\text{ s}$
    • B) $T = 1.90\text{ s}$ with $\Delta T = \pm 8.00\text{ s}$
    • C) $T = 1.90\text{ s}$ with $\Delta T = \pm 0.20\text{ s}$
    • D) $T = 1.90\text{ s}$ with $\Delta T = \pm 0.02\text{ s}$
16. 16.

    During a laboratory investigation, a student records five repeated measurements for the time of flight of a projectile: $3.42\text{ s}$, $3.46\text{ s}$, $3.40\text{ s}$, $3.48\text{ s}$, and $3.44\text{ s}$. According to standard experimental practice, the measured value is reported as the arithmetic mean, and the absolute random uncertainty is estimated as half the range of the data set. Which expression correctly reports the time of flight in the standard form $(\bar{t} \pm \Delta t)$?

    • A) $(3.44 \pm 0.04)\text{ s}$
    • B) $(3.44 \pm 0.08)\text{ s}$
    • C) $(3.44 \pm 0.02)\text{ s}$
    • D) $(3.44 \pm 0.01)\text{ s}$
17. 17.

    In an experiment to determine the electrical resistance of a component using the relationship $R = \frac{V}{I}$, a student records a voltage of $(12.0 \pm 0.6)\text{ V}$ and a current of $(2.00 \pm 0.02)\text{ A}$. The student wants to evaluate the reliability of the calculated resistance and determine which measurement should be refined to reduce the overall uncertainty most effectively. What is the percentage uncertainty in the calculated resistance, and which measurement should the student prioritize improving?

    • A) 4%, and the student should prioritize improving the voltage measurement.
    • B) 5%, and the student should prioritize improving the current measurement.
    • C) 6%, and the student should prioritize improving the voltage measurement.
    • D) 6%, and the student should prioritize improving the current measurement.
18. 18.

    An experimenter measures the mass of a metal block using an electronic balance that consistently reads +0.50 g when empty. Repeated measurements of the same block in an open room yield values between 42.10 g and 42.30 g due to fluctuating air currents. Which statement correctly analyzes these sources of error and the proper method to determine an accurate value for the block's mass?

    • A) The zero offset is a random error and the air currents cause a systematic error; calculating the mean of repeated trials completely removes both errors.
    • B) Both factors cause systematic errors that reduce accuracy; neither error can be reduced by taking repeated measurements or shielding the balance.
    • C) The zero offset is a systematic error that reduces precision, while air currents cause random errors that reduce accuracy; shielding the balance eliminates both errors without any numerical correction.
    • D) The zero offset is a systematic error affecting accuracy, while air currents cause random errors affecting precision; the air current effect is reduced by averaging repeated trials, and the zero offset must be subtracted from this average.
19. 19.

    During an electrical resistivity experiment, a student measures the diameter of a uniform metal wire using a micrometer screw gauge and records the value as $d = (5.00 \pm 0.15)\text{ mm}$. To determine the resistivity, the student must first compute the cross-sectional area of the wire using the relationship $A = \frac{\pi d^2}{4}$. Based on the principles of uncertainty propagation, what is the percentage uncertainty in the calculated cross-sectional area?

    • A) 1.5%
    • B) 3.0%
    • C) 6.0%
    • D) 9.0%
20. 20.

    A student measures the mass of an empty beaker as $(125.0 \pm 0.5)\text{ g}$ and the mass of the same beaker filled with a liquid sample as $(375.0 \pm 0.5)\text{ g}$. The student needs to determine the net mass of the liquid sample along with its associated measurement uncertainty. What are the absolute uncertainty and the percentage uncertainty of the calculated liquid mass?

    • A) $\pm 0.0\text{ g}$ and $0.0\%$
    • B) $\pm 0.5\text{ g}$ and $0.2\%$
    • C) $\pm 1.0\text{ g}$ and $0.2\%$
    • D) $\pm 1.0\text{ g}$ and $0.4\%$

Answers and step-by-step solutions appear after you finish the test.