Exploring & Designing - IB Physics

A student is testing the hypothesis that the kinetic energy ( $E_k$ ) of an object is proportional to the square of its velocity ( $v$ ). They measure $E_k$ and $v$ for an object of constant mass. To confirm the hypothesis using a linear graph, which quantities should they plot on the y-axis and x-axis respectively?
Answer: $E_k$ on the y-axis and $v^2$ on the x-axis. The relationship is $E_k = \frac{1}{2}mv^2$ . To linearize this equation in the form $y = mx + c$ , we can treat $E_k$ as $y$ and $v^2$ as $x$ . The equation becomes $y = (\frac{1}{2}m)x$ . This shows that a graph of $E_k$ versus $v^2$ will be a straight line passing through the origin with a gradient of $\frac{1}{2}m$ .
A student measures the dimensions of a rectangular block to calculate its volume. The measurements are length $L = 50.0 \pm 0.5$ cm, width $W = 10.0 \pm 0.5$ cm, and height $H = 2.0 \pm 0.5$ cm. Which measurement is the largest source of uncertainty in the calculated volume?
Answer: The height, H, because it has the largest percentage uncertainty. When multiplying quantities, the percentage (or fractional) uncertainties add up. To find the largest source of uncertainty, we must compare the percentage uncertainties of each measurement. % unc(L) = (0.5/50.0) 100% = 1%. % unc(W) = (0.5/10.0) 100% = 5%. % unc(H) = (0.5/2.0) * 100% = 25%. The height measurement has the largest percentage uncertainty (25%) and will therefore contribute the most to the uncertainty in the final calculated volume.
When designing an experiment to measure the Young's modulus of a wire using Hooke's law, a student must choose between two approaches: (I) applying increasing loads and measuring total extension, or (II) applying and removing loads cyclically while measuring extension. Which approach is preferable and why?
Answer: Approach II, because it allows detection of plastic deformation and ensures measurements stay within the elastic limit Young's modulus ( $E = \frac{\sigma}{\epsilon} = \frac{FL_0}{A\Delta L}$ ) is only valid in the elastic region where Hooke's law applies. Approach II (cyclic loading) reveals whether the material returns to its original length, indicating elastic behavior. If plastic deformation occurs, the unloading curve differs from loading, warning that the elastic limit was exceeded. Choice A overlooks the fundamental requirement of staying elastic. Choice C incorrectly assumes significant fatigue in a few cycles. Choice D mentions error reduction but misses the primary benefit of detecting plastic deformation.
When designing an experiment to measure the specific heat capacity of an unknown metal using the method of mixtures, which factor would most significantly compromise the validity of the results if not properly controlled?
Answer: Failing to account for the heat capacity of the calorimeter and stirrer in the energy balance calculations The calorimeter and stirrer absorb significant thermal energy during the mixing process. Ignoring their heat capacity introduces a systematic error that makes the calculated specific heat capacity consistently too low, as the energy balance equation $m_m c_m \Delta T_m = (m_w c_w + C_{cal}) \Delta T_w$ requires the calorimeter heat capacity $C_{cal}$ . Choice A is actually better practice (distilled water has known properties). Choice C affects precision but not validity if applied consistently. Choice D is acceptable experimental practice as long as masses are recorded.
A student is designing an experiment to investigate the interference of sound waves using two loudspeakers connected to the same signal generator. To observe a stable interference pattern of maxima and minima, what is the most critical condition that must be met by the two sound sources?
Answer: The loudspeakers must be coherent, meaning they emit waves with a constant phase difference. While other factors are important for a clear pattern, the fundamental requirement for a stable, observable interference pattern is that the sources must be coherent. Coherence (a constant phase relationship) ensures that the positions of constructive interference (maxima) and destructive interference (minima) do not shift over time. Connecting both speakers to the same signal generator achieves this. Same frequency (B) is necessary for coherence, but coherence is the more precise and encompassing term for the required condition.
A student designs an experiment to investigate how the period of a simple pendulum depends on amplitude. The student plans to measure periods for amplitudes of 5°, 15°, 30°, 45°, and 60°. Which modification would most improve the experimental design?
Answer: Include additional amplitude measurements at 2°, 8°, and 12° to better resolve the small-amplitude region where theory predicts linear behavior The simple harmonic motion approximation ( $T = 2\pi\sqrt{L/g}$ ) is most accurate for small amplitudes (typically <15°). To investigate how period depends on amplitude, the student needs data points in the small-amplitude region where the approximation should hold, and larger amplitudes where deviations appear. Choice A provides better resolution where the transition occurs. Choice B tests extreme cases but loses focus on the theoretically important small-amplitude region. Choice C reduces data density. Choice D changes the research question entirely.
A student measures the length of an object with a ruler. They repeat the measurement five times and get the following readings: 25.4 cm, 25.5 cm, 25.3 cm, 25.5 cm, 25.4 cm. Later, they discover the ruler had a zero error, with its end at the -0.2 cm mark instead of 0.0 cm. How should the errors in this experiment be described?
Answer: There are both random errors causing the scatter and a systematic error causing inaccuracy. The variation in the readings (e.g., 25.3, 25.4, 25.5) indicates the presence of random errors, which cause scatter around an average value. The zero error on the ruler, which causes every reading to be consistently off by the same amount (in this case, 0.2 cm too high), is a systematic error. Real experiments almost always contain both types of error.
A student hypothesizes that the resistance $R$ of a thermistor is related to its absolute temperature $T$ by the equation $R = R_0 e^{B/T}$ , where $R_0$ and $B$ are constants. To obtain a straight-line graph from which the constant $B$ can be determined, which quantities should be plotted?
Answer: $\ln(R)$ on the y-axis and $1/T$ on the x-axis. To linearize the exponential equation $R = R_0 e^{B/T}$ , we can take the natural logarithm of both sides: $\ln(R) = \ln(R_0 e^{B/T})$ . Using logarithm rules, this becomes $\ln(R) = \ln(R_0) + \ln(e^{B/T})$ , which simplifies to $\ln(R) = B(1/T) + \ln(R_0)$ . This equation is in the form $y = mx + c$ , where $y = \ln(R)$ , $x = 1/T$ , the gradient $m = B$ , and the y-intercept $c = \ln(R_0)$ . Therefore, plotting $\ln(R)$ versus $1/T$ will yield a straight line with a gradient equal to B.
A student wants to investigate the relationship between the angle of incidence and the angle of refraction when light passes from air into glass. Which combination of variables and controls would provide the most reliable data for determining Snell's law?
Answer: Measure multiple angles of incidence with the same wavelength of light, keeping the glass thickness and surface quality constant To determine Snell's law ( $n_1 \sin \theta_1 = n_2 \sin \theta_2$ ), the student needs to establish the relationship between angle of incidence and angle of refraction. This requires varying the independent variable (angle of incidence) while keeping all other factors constant (wavelength, glass type, surface quality). Glass thickness doesn't affect refraction at the air-glass interface. Choice B tests dispersion, not Snell's law. Choice C would test how refractive index varies with material. Choice D introduces unnecessary variables that could obscure the fundamental relationship.
To determine the thickness of a single sheet of aluminium foil as precisely as possible using only a 30 cm ruler and a digital balance, which is the most appropriate method?
Answer: Measure the length and width of a single sheet, measure its mass, and calculate thickness using the known density of aluminium. A single sheet of foil is far too thin to be measured directly by a ruler. Stacking (A) is difficult with foil, and folding (C) introduces large errors. Water displacement (D) is also impractical for a thin sheet. The best method is indirect: measure the macroscopic properties of a single large sheet, its length and width (to find area $A$ ) and its mass $m$ . Using the known value for the density of aluminium ( $\rho$ ) from a data booklet, the volume can be found ( $V=m/\rho$ ). Since $V = A \times \text{thickness}$ , the thickness can be calculated as $t = V/A = m/(\rho A)$ .
A student designs an experiment to measure the speed of sound in air by analyzing the resonance frequencies of a tube closed at one end. The tube length can be adjusted, and a speaker provides sound waves. What is the most significant experimental limitation that could affect the accuracy of the speed calculation?
Answer: The finite diameter of the tube causes the effective length to differ from the measured length by an end correction factor When analyzing resonance in closed tubes, you're measuring standing wave patterns where the tube length relates to specific wavelengths. The fundamental resonance occurs when the tube length equals one-quarter wavelength, with subsequent resonances at odd multiples. Using the relationship $v = f\lambda$ , you can calculate sound speed from the frequency and wavelength data. The most significant limitation is D - the finite diameter creates an end correction that makes the effective acoustic length longer than the physical tube length. At the open end, the sound wave extends slightly beyond the tube opening (typically about 0.6 times the radius), meaning your measured length systematically underestimates the true acoustic length. This creates a consistent error in all wavelength calculations, directly affecting your speed calculation. A is incorrect because background noise, while annoying, doesn't systematically bias your length measurements - you can still identify resonance peaks with careful observation. B represents a real but minor effect since temperature changes during a typical lab session cause only small speed variations (about 0.6 m/s per degree Celsius). C is wrong because harmonics actually help confirm resonance conditions rather than obscure them - multiple frequency components will all show enhanced amplitude at true resonance. Study tip: In IB Physics practicals involving wave measurements, always consider geometric factors that affect the relationship between measured and effective lengths. End corrections are a classic source of systematic error in acoustic experiments, so mention this limitation when discussing experimental uncertainties.
A student plots a graph of current $I$ on the y-axis against potential difference $V$ on the x-axis for a metallic conductor at constant temperature. They find the gradient of the resulting straight-line graph. What physical quantity does this gradient represent?
Answer: The reciprocal of the resistance $1/R$ of the conductor. Ohm's Law is given by $V=IR$ . The student is plotting a graph of $y=I$ against $x=V$ . To match the form of a linear equation $y=mx+c$ , we rearrange Ohm's Law to make $I$ the subject: $I = (1/R)V$ . In this form, the gradient $m$ corresponds to $1/R$ , the reciprocal of the resistance (also known as conductance).
A student investigating the photoelectric effect plans to measure the maximum kinetic energy of photoelectrons using different wavelengths of incident light. Which experimental consideration is most critical for obtaining reliable results that can verify Einstein's photoelectric equation?
Answer: Ensuring the metal surface is clean and unoxidized, and using monochromatic light sources with precisely known wavelengths Einstein's equation $KE_{max} = hf - \phi$ requires accurate measurement of photon energy ( $hf = hc/\lambda$ ) and maximum kinetic energy. Surface contamination changes the effective work function, and non-monochromatic light makes photon energy uncertain. Choice B misunderstands that intensity affects the number of photoelectrons, not their maximum energy. Choice C is less critical since the work function temperature dependence is usually small over reasonable temperature ranges. Choice D tests a different aspect but isn't necessary for verifying the basic equation.
A student wants to investigate the relationship between the frequency of a vibrating string and its tension. The experimental setup includes strings of identical material and length. Which sequence of measurements would provide the most reliable data for this investigation?
Answer: Alternate between increasing tension from 10N to 50N and decreasing from 50N to 10N, recording frequencies throughout The relationship $f = \frac{1}{2L}\sqrt{\frac{T}{\mu}}$ suggests frequency should increase with tension. However, strings can exhibit hysteresis, and the setup might drift over time. Approach D tests for systematic effects by checking whether the relationship is the same during increasing and decreasing tension phases, while also spreading measurements over time. Choice A concentrates all measurements of each tension together, making it sensitive to temporal drift. Choice B has the same problem. Choice C loses the systematic exploration of the relationship and makes trend analysis difficult.
An investigation aims to determine the relationship between the power dissipated in a resistor and the applied voltage. A student measures current and voltage across various resistors while keeping each resistor at constant temperature. Which analysis approach would most effectively distinguish between $P \propto V$ and $P \propto V^2$ ?
Answer: Plot $\log(P)$ versus $\log(V)$ and determine the slope; a slope of 1 indicates $P \propto V$ , while a slope of 2 indicates $P \propto V^2$ For a power law relationship $P = kV^n$ , taking logarithms gives $\log(P) = \log(k) + n\log(V)$ . A log-log plot yields a straight line with slope $n$ . This method directly determines the exponent and works well even with experimental scatter. Choice A relies on subjective visual assessment and may be ambiguous. Choice B measures correlation strength but both relationships could have high correlations. Choice D compares specific functional forms but doesn't directly determine the power law exponent, and the quadratic fit assumes $P \propto V^2$ rather than testing it.
A student investigates the rate of cooling of a liquid in two different containers, A and B. Container A has a dull black surface, and container B has a shiny silver surface. Both contain the same volume of the same liquid at the same initial temperature. The student concludes that the liquid in container A cools faster because dull black surfaces are better emitters of thermal radiation. Why is this conclusion potentially incomplete?
Answer: The material of container A might be a better thermal conductor than container B. While it is true that dull black surfaces are better emitters (and absorbers) of radiation, heat can also be lost through conduction and convection. If container A is made of a material with a much higher thermal conductivity than container B, it would lose heat faster via conduction through its base and sides. This is a confounding variable. The student's conclusion attributes the entire effect to radiation, which may be invalid if conduction is also a significant, uncontrolled factor.
A student is designing an experiment to measure the half-life of a radioactive sample. A key part of the procedure is accounting for background radiation. What is the correct procedure for doing this?
Answer: Measure the background count rate over a long period before and after the experiment and use the average. Background radiation is random and can fluctuate. To get a reliable average value, it should be measured for a time period comparable to the main experiment's duration. Measuring it before and after and taking an average accounts for any potential slow drifts in background activity. This average background rate is then subtracted from each measurement of the source's count rate to find the corrected activity of the source itself.
When investigating Hooke's Law, a student plots a graph of applied force versus spring extension. The data points form a clear straight line that is offset from the origin. Which experimental error is the most likely cause for the best-fit line not passing through the origin?
Answer: Failing to account for the mass of the weight hanger when recording the applied force. If the mass of the weight hanger is not accounted for, every force value recorded ( $F = m_{added}g$ ) will be systematically lower than the actual force acting on the spring ( $F_{actual} = (m_{added} + m_{hanger})g$ ). This adds a constant offset to the force, shifting the entire graph. For example, when $m_{added}=0$ , there is still a force from the hanger, causing an extension. This means the graph will not pass through the (0,0) origin. Random errors (D) cause scatter, and exceeding the elastic limit (C) would make the graph non-linear.
A student conducts an experiment to measure the specific heat capacity of a metal block by heating it and observing the temperature rise. To ensure the thermal energy supplied is accurately calculated, they use a voltmeter and an ammeter connected to the heater. What is the correct placement of these meters in the circuit?
Answer: The ammeter should be in series with the heater, and the voltmeter should be in parallel with the heater. To measure the power $P=IV$ delivered to the heater, one must measure the current $I$ flowing through it and the potential difference $V$ across it. Ammeters measure current and have very low resistance, so they must be connected in series to have the circuit current flow through them. Voltmeters measure potential difference and have very high resistance, so they must be connected in parallel across the component to measure the potential drop without drawing significant current from the main circuit.
An experiment investigates how the magnetic field strength at the center of a solenoid depends on the current. The student plans to use solenoids with different numbers of turns per unit length. Which approach would best isolate the current dependence?
Answer: Use multiple solenoids with different turn densities, plot $B/n$ versus current for each, and verify all data collapse to the same line The theoretical relationship $B = \mu_0 nI$ predicts that $B/n$ should be proportional to current with the same proportionality constant ( $\mu_0$ ) for all solenoids. Plotting $B/n$ vs $I$ for different turn densities tests whether all data collapse to a single line, providing strong evidence for the theoretical relationship. Choice A loses information by averaging. Choice B uses limited data from one solenoid. Choice D requires comparing slopes, which is less definitive than demonstrating data collapse and doesn't directly test the functional form.

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