Measurements and uncertainties：练习题

An experiment is performed to determine the acceleration of free fall \(g\) by measuring the time \(t\) taken for a ball to fall through a height \(h\). The equation used is \(g = \frac{2h}{t^2}\). The percentage uncertainty in \(h\) is \(2\%\) and the percentage uncertainty in \(t\) is \(3\%\). What is the percentage uncertainty in the calculated value of \(g\)?

The density \(\rho\) of a sphere is determined by measuring its mass \(M\) and its diameter \(d\). The uncertainty in \(M\) is \(1\%\) and the uncertainty in \(d\) is \(2\%\). What is the percentage uncertainty in the calculated density?

A student uses a micrometer screw gauge to measure the diameter of a wire. The micrometer has a zero error of \(-0.02\text{ mm}\). The student records a reading of \(1.56\text{ mm}\). The manufacturer states the micrometer has an absolute uncertainty of \(\pm 0.01\text{ mm}\). What is the corrected diameter and its percentage uncertainty?

The power \(P\) dissipated in a resistor is given by \(P = \frac{V^2}{R}\). The potential difference \(V\) is measured as \((12.0 \pm 0.3)\text{ V}\) and the resistance \(R\) is measured as \((6.0 \pm 0.3)\text{ }\Omega\). What is the value of \(P\) with its absolute uncertainty?

A student measures the period \(T\) of a simple pendulum by timing \(20\) oscillations with a stopwatch. The total time for \(20\) oscillations is \(32.4 \pm 0.2\text{ s}\). The length of the pendulum is measured as \(65.0 \pm 0.5\text{ cm}\). If \(g = \frac{4\pi^2 L}{T^2}\), what is the percentage uncertainty in the experimental value of \(g\)?

A student uses a micrometer to measure the diameter of a thin wire. The readings obtained are \(0.22\text{ mm}\), \(0.21\text{ mm}\), \(0.22\text{ mm}\), and \(0.23\text{ mm}\). Calculate the percentage uncertainty in the average diameter.

The power \(P\) dissipated in a resistor is calculated using \(P = I^2 R\). If the uncertainty in the current \(I\) is \(3\%\) and the uncertainty in the resistance \(R\) is \(5\%\), calculate the absolute uncertainty in the power when the measured values are \(I = 2.0 \pm 0.06\text{ A}\) and \(R = 10.0 \pm 0.5\text{ }\Omega\).

The period \(T\) of a simple pendulum is given by \(T = 2\pi\sqrt{\frac{L}{g}}\). If the percentage uncertainty in the length \(L\) is \(2\%\) and the percentage uncertainty in the period \(T\) is \(3\%\), determine the percentage uncertainty in the calculated value of the acceleration of free fall \(g\).

A student conducts an experiment to determine the acceleration of free fall \( g \) by dropping a heavy metal ball from rest. The distance fallen \( h \) and the time taken \( t \) are recorded. The student plots a graph of \( h \) on the y-axis against \( t^2 \) on the x-axis.
(a) Explain why the gradient of this graph is equal to \( \frac{1}{2}g \).
(b) The student records \( h = 1.500 \pm 0.002 \text{ m} \) and \( t = 0.55 \pm 0.02 \text{ s} \). Calculate the percentage uncertainty in the value of \( g \) derived from these single measurements.
(c) Suggest why a graph of \( h \) against \( t^2 \) is more accurate for determining \( g \) than a single calculation from one set of measurements.