Interpreting Results: notes and practice questions

Two students, X and Y, are measuring the period of rotation, $T$, of a turntable using a digital stopwatch with a precision of $0.01 \text{ s}$. The turntable is set to rotate at a constant angular velocity.

Student X measures the time for one single, complete revolution. They repeat this measurement 20 times and then calculate the mean of their readings.

Student Y measures the total time taken for 20 consecutive revolutions and then divides this total time by 20 to find the period.

(a) The stopwatch is functioning correctly. State the principal source of random uncertainty in an individual time measurement.

(b) Suggest a reasonable value for the absolute uncertainty in a single time measurement.

(c) Explain which student's method will yield a more precise value for the period $T$.

A student conducts an experiment to investigate the relationship between the period $T$ of a simple pendulum and its length $L$. The student measures the time taken for 20 complete oscillations for various lengths of the pendulum string.

(a) State:

(i) the independent variable.

(ii) the dependent variable.

(iii) one variable that should be controlled.

(b) The student uses a digital stopwatch to measure the time. They have a consistent reaction time, starting the stopwatch 0.15 s after the pendulum bob is released from its highest point. State and explain whether this is a random or a systematic error.

(c) The student measures the time for 20 oscillations rather than for a single oscillation. Suggest why this experimental technique improves the measurement of the period $T$.

(d) For a pendulum of length 1.00 m, the accepted value for the period is 2.01 s. The student takes three measurements of the time for 20 oscillations and calculates the period for each. The results are 2.15 s, 2.16 s, and 2.14 s. Distinguish between accuracy and precision, using the student's results as an example.

A student performs an experiment to determine the resistivity of a metal wire. The resistance $R$, length $L$, and diameter $d$ of the wire are measured.

(a) The diameter is measured with a micrometer screw gauge. State one experimental precaution that should be taken to ensure this measurement is accurate.

(b) The following data are collected:

Resistance $R = (5.2 \pm 0.1) \, \Omega$

Length $L = (1.50 \pm 0.01) \, \text{m}$

Diameter $d = (0.45 \pm 0.02) \, \text{mm}$

Calculate the resistivity of the wire and its absolute uncertainty.

(c) State the value of the resistivity with its uncertainty to an appropriate number of significant figures.

(d) The accepted value for the resistivity of nichrome at the experimental temperature is $1.10 \times 10^{-6} \, \Omega \text{ m}$. Deduce, using your answer from (c), whether the wire is likely to be made of nichrome.

A student investigates the cooling of a hot liquid. A cup containing hot water is placed in a room where the ambient temperature is constant. The temperature $T$ of the water is measured at regular time intervals $t$.

The data collected is shown in the table below:

It is observed that the rate of cooling decreases as the temperature of the water approaches the ambient temperature. The student suggests that the cooling process can be modelled by an exponential decay, similar to radioactive decay, where the temperature difference $\Delta T = T - T_a$ (where $T_a$ is the ambient temperature) decreases exponentially with time.

(a) The ambient temperature in the room is $20.0$ °C. Estimate the half-cooling time of the water, which is the time taken for the temperature difference $\Delta T$ to halve.

(b) Show that the cooling process follows an exponential decay model by calculating the decay constant $k$ from two different time intervals and comparing the values. State your chosen time intervals clearly.

The relationship for exponential decay is given by $\Delta T(t) = \Delta T_0 e^{-kt}$, where $\Delta T_0$ is the initial temperature difference.

(c) The student wants to stop the experiment when the temperature of the water has dropped to $30.0$ °C. Predict the time at which the student will stop the experiment.

A group of engineers is designing a new rectangular solar panel. They need to accurately determine its surface area. The dimensions of the panel are measured as length $L = (1.60 \pm 0.01)$ m and width $W = (0.95 \pm 0.01)$ m. The thickness of the panel's protective glass layer is measured as $t = (4.50 \pm 0.05)$ mm.

(a) Suggest an appropriate measuring instrument for determining the thickness $t$.

(b) Calculate the percentage uncertainty in the calculated surface area of the solar panel.