Limitation of physical measurements：練習問題

A student uses different instruments to measure the dimensions of a small metal block. Which of the following instruments would provide the highest resolution for a measurement of approximately \(15 \, \text{mm}\)?

A toy car of mass \(m\) moves at a velocity \(v\). The percentage uncertainty in the measurement of \(m\) is \(2\%\) and the percentage uncertainty in the measurement of \(v\) is \(4\%\). What is the percentage uncertainty in the calculated kinetic energy \(E_k\)?

In an experiment to determine the spring constant \(k\), a student plots a graph of force \(F\) against extension \(\Delta L\). The best-fit line has a gradient of \(40.0 \, \text{N m}^{-1}\). The student determines that the maximum possible gradient (steepest line) passing through the error bars is \(41.5 \, \text{N m}^{-1}\) and the minimum possible gradient (shallowest line) is \(37.5 \, \text{N m}^{-1}\). What is the value of \(k\) with its associated absolute uncertainty?

A student repeats an experiment to find the acceleration due to gravity \(g\). The values obtained are \(10.21 \, \text{m s}^{-2}\), \(10.19 \, \text{m s}^{-2}\), and \(10.20 \, \text{m s}^{-2}\). The accepted value for \(g\) is \(9.81 \, \text{m s}^{-2}\). Which statement best describes these results?

A rectangular plate has a measured length \(L = (10.0 \pm 0.2) \, \text{cm}\) and width \(W = (5.0 \pm 0.1) \, \text{cm}\). If the area \(A = L \times W\), what is the absolute uncertainty in the calculated area \(A\)?

The current flowing through a resistor is measured as \((2.0 \pm 0.1) \text{ A}\) and the potential difference across it is \((12.0 \pm 0.5) \text{ V}\). Calculate the percentage uncertainty in the calculated resistance \(R\).

A sphere has a diameter measured as \(D = (4.0 \pm 0.2) \text{ cm}\). Calculate the percentage uncertainty in the volume \(V\) of the sphere, assuming \(V \propto D^3\).

A student measures the time period of a pendulum and finds consistent values that are all slightly too long due to a slow reaction time in starting the stopwatch. Identify the type of error and suggest one way to minimize it.

A student measures the temperature of water at the start and end of a heating experiment using a thermometer with a resolution of \(0.1 \, ^\circ\text{C}\). The readings taken are:

• Initial temperature, \(T_1 = (20.5 \pm 0.1) \, ^\circ\text{C}\)
• Final temperature, \(T_2 = (85.2 \pm 0.1) \, ^\circ\text{C}\)

(a) Calculate the temperature rise \(\Delta T\).

(b) Determine the absolute uncertainty in the temperature rise \(\Delta T\).

(c) Calculate the percentage uncertainty in the temperature rise \(\Delta T\).

A student determines the density \(\rho\) of a uniform metal cylinder using the relationship \(\rho = \frac{M}{V}\), where \(V = \pi r^2 L\). The following measurements are recorded:

• Mass, \(M = (150.0 \pm 0.5) \, \text{g}\)
• Radius, \(r = (1.00 \pm 0.02) \, \text{cm}\)
• Length, \(L = (5.0 \pm 0.1) \, \text{cm}\)

(a) Calculate the density \(\rho\) in \(\text{g cm}^{-3}\).

(b) Calculate the fractional uncertainty in the volume \(V\).

(c) Calculate the absolute uncertainty in the density \(\rho\), expressing your final answer for density with its associated uncertainty to an appropriate number of significant figures.