物理測量的限制與誤差：練習題

A student repeats a set of measurements for the time period of a pendulum. If the student obtains the same result using the same equipment and method, the measurement is described as:

A student measures the diameter of a wire as \( 0.52 \text{ mm} \) with an absolute uncertainty of \( 0.02 \text{ mm} \). Which of the following correctly expresses this measurement to the appropriate number of significant figures consistent with its uncertainty?

A student plots a graph of the square of the period \( T^2 \) against the length \( L \) of a simple pendulum to determine the acceleration due to gravity, \( g \). The gradient of the line of best fit is \( 4.02 \text{ s}^2 \text{ m}^{-1} \). The line of steepest possible fit has a gradient of \( 4.14 \text{ s}^2 \text{ m}^{-1} \) and the line of shallowest possible fit has a gradient of \( 3.94 \text{ s}^2 \text{ m}^{-1} \). Given the relationship \( T^2 = \frac{4\pi^2}{g} L \), what is the absolute uncertainty in the calculated value of \( g \)?

A student uses a stopwatch to measure the time for 10 oscillations of a pendulum. The results are as follows:
Trial 1: 15.6 s
Trial 2: 15.4 s
Trial 3: 15.5 s
Trial 4: 15.9 s

What is the absolute uncertainty in the mean time for 10 oscillations based on the range of these measurements?

A student measures the mass of an object as \( m = (250 \pm 5) \text{ g} \) and its volume as \( V = (100 \pm 2) \text{ cm}^3 \). Calculate the percentage uncertainty in the density of the object.

Define the term resolution as it applies to a measuring instrument like a digital voltmeter.

A rectangular plate has a length measured as \(20.0 \pm 0.2\text{ cm}\) and a width of \(10.0 \pm 0.1\text{ cm}\). Calculate the percentage uncertainty in the calculated area of the plate.

A researcher plots a graph of \( y \) against \( x \) to find a physical constant from the gradient. The line of best fit has a gradient of \( 4.50 \text{ units} \). The student then draws a maximum possible gradient line (steepest worst fit) passing through the error bars with a value of \( 4.72 \text{ units} \), and a minimum possible gradient line of \( 4.34 \text{ units} \). Calculate the absolute uncertainty in the gradient and express the final gradient value with its uncertainty to the correct number of significant figures.

A student plots a graph of force \( F \) against extension \( \Delta L \) for a spring to determine the spring constant \( k \). One specific data point is recorded as \( F = 6.0 \pm 0.2 \text{ N} \) and \( \Delta L = 15.0 \pm 0.5 \text{ mm} \).
(a) Calculate the percentage uncertainty for both the force and the extension measurements.
(b) Determine the percentage uncertainty in the calculated spring constant \( k \), where \( k = \frac{F}{\Delta L} \).
(c) Describe how the uncertainties in \( F \) and \( \Delta L \) would be represented visually on the graph using error bars.

A physicist is conducting an experiment to measure the Planck constant \( h \) using the photoelectric effect. The stopping potential \( V_s \) is measured for various frequencies \( f \) of incident light. The data is plotted on a graph of \( V_s \) against \( f \).

(a) The gradient of the line of best fit is calculated as \( 4.10 \times 10^{-15} \text{ V s} \). The line of steepest possible fit has a gradient of \( 4.25 \times 10^{-15} \text{ V s} \) and the line of shallowest possible fit has a gradient of \( 3.95 \times 10^{-15} \text{ V s} \). Calculate the absolute uncertainty in the gradient.
(b) Using the equation \( eV_s = hf - \phi \), where \( e = 1.60 \times 10^{-19} \text{ C} \), determine the value of \( h \) and its absolute uncertainty. Give your answer to an appropriate number of significant figures.
(c) The y-intercept of the graph is found to be \( -1.80 \pm 0.15 \text{ V} \). Calculate the work function \( \phi \) in electronvolts (\( \text{eV} \)) and its percentage uncertainty.