Linearizing data Transitional Reasoning Download PDF

Questions

1. 1.

   In an investigation of uniform circular motion, a student measures the tension force $F$ required to keep a puck of mass $m = 0.20\text{ kg}$ moving in a horizontal circle of constant radius $r$ at various speeds $v$. According to theoretical mechanics, the variables satisfy $F = \left(\frac{m}{r}\right)v^2$. To linearize the non-linear data, the student plots $F$ on the vertical axis against $v^2$ on the horizontal axis, obtaining a straight best-fit line with a slope of $0.80\text{ kg/m}$. Based on this linear fit, what is the radius $r$ of the circular path, and what would the slope be if the experiment were repeated with the same radius using a puck of mass $0.50\text{ kg}$?

   • A) $r = 0.16\text{ m}$, and the new slope is $0.32\text{ kg/m}$
   • B) $r = 0.25\text{ m}$, and the new slope is $2.0\text{ kg/m}$
   • C) $r = 0.25\text{ m}$, and the new slope is $0.32\text{ kg/m}$
   • D) $r = 4.0\text{ m}$, and the new slope is $0.13\text{ kg/m}$
2. 2.

   In an investigation of mechanical waves, a student measures the speed $v$ of transverse waves along two different stretched cords, Cord 1 and Cord 2, under various tension forces $F$. According to wave theory, the speed is governed by $v = \sqrt{\frac{F}{\mu}}$, where $\mu$ is the linear mass density of the cord. To linearize the data, the student plots $v^2$ on the vertical axis against $F$ on the horizontal axis for each cord and fits a straight line passing through the origin. If the slope of the best-fit line for Cord 1 is 4 times the slope of the best-fit line for Cord 2, which statement correctly compares the linear mass densities of the two cords?

   • A) The linear mass density of Cord 1 is $\frac{1}{4}$ that of Cord 2 because the slope of the line is inversely proportional to $\mu$.
   • B) The linear mass density of Cord 1 is 4 times that of Cord 2 because the slope of the line is directly proportional to $\mu$.
   • C) The linear mass density of Cord 1 is $\frac{1}{2}$ that of Cord 2 because the wave speed varies inversely with the square root of $\mu$.
   • D) The linear mass density of Cord 1 is 2 times that of Cord 2 because the slope of the line is directly proportional to $\sqrt{\mu}$.
3. 3.

   In an investigation of an enclosed ideal gas held at a constant temperature, a student measures the absolute pressure $P$ as the volume $V$ is varied. According to the ideal gas law, the quantities follow the relationship $P = \frac{nRT}{V}$, where $n$ is the amount of gas, $R$ is the universal gas constant, and $T$ is the absolute temperature. To linearize the non-linear data, the student plots $P$ on the vertical axis against $\frac{1}{V}$ on the horizontal axis and fits a straight line passing through the origin. Which statement correctly identifies the physical meaning of the slope $S$ of this best-fit line and predicts how $S$ changes if the experiment is repeated with the same gas sample at a higher absolute temperature?

   • A) The slope $S$ represents $\frac{nR}{T}$, and the slope will decrease because temperature appears in the denominator of the rate expression.
   • B) The slope $S$ represents $nRT$, and the slope will increase because $S$ is directly proportional to absolute temperature.
   • C) The slope $S$ represents $\frac{1}{nRT}$, and the slope will remain unchanged because the gas constant $R$ is invariant.
   • D) The slope $S$ represents $nRT$, and the slope will decrease because higher temperatures cause the gas to expand to greater volumes.
4. 4.

   In a photoelectric effect experiment, a student measures the stopping potential $V_s$ for light of various wavelengths $\lambda$ incident on a target metal. According to Einstein's photoelectric model, the quantities follow the relationship $V_s = \left(\frac{hc}{e}\right)\frac{1}{\lambda} - \frac{\Phi}{e}$, where $h$ is Planck's constant, $c$ is the speed of light in vacuum, $e$ is the elementary charge, and $\Phi$ is the work function. To linearize the non-linear data, the student plots $V_s$ on the vertical axis against $\frac{1}{\lambda}$ on the horizontal axis, obtaining a straight best-fit line with slope $M$ and horizontal intercept $x_0$. Which pair of expressions correctly determines Planck's constant $h$ and the threshold wavelength $\lambda_0$ of the metal from this linear fit?

   • A) $h = \frac{Mc}{e}$ and $\lambda_0 = x_0$
   • B) $h = \frac{eM}{c}$ and $\lambda_0 = \frac{1}{x_0}$
   • C) $h = \frac{eM}{c}$ and $\lambda_0 = x_0$
   • D) $h = \frac{M}{ec}$ and $\lambda_0 = \frac{1}{x_0}$
5. 5.

   In an experiment investigating acoustic resonance, a student measures the tube length $L$ at which the fundamental resonance occurs for several tuning forks of known frequency $f$. The theoretical relationship between the variables is given by $L + c = \frac{v}{4f}$, where $v$ is the speed of sound in air and $c$ is the tube's acoustic end correction. To linearize the non-linear data, the student plots $L$ on the vertical axis against $\frac{1}{f}$ on the horizontal axis and obtains a straight best-fit line with slope $m$ and vertical intercept $b$. Which pair of expressions correctly determines the speed of sound $v$ and the end correction $c$ from the linear fit parameters?

   • A) $v = 4m$ and $c = -b$
   • B) $v = \frac{m}{4}$ and $c = b$
   • C) $v = 4m$ and $c = b$
   • D) $v = \frac{4}{m}$ and $c = -b$
6. 6.

   In an experiment studying the discharge of a capacitor through a resistor, the potential difference $V$ across the capacitor decreases over time $t$ according to the relation $V = V_0 e^{-t/\tau}$, where $V_0$ is the initial potential difference and $\tau$ is the circuit time constant. To linearize this relationship, a student takes the natural logarithm of both sides to obtain $\ln V = -\frac{1}{\tau}t + \ln V_0$ and plots $\ln V$ on the vertical axis versus $t$ on the horizontal axis. If the resulting best-fit line has a slope $m$ and a vertical intercept $b$, which pair of expressions correctly determines $\tau$ and $V_0$ from the graph?

   • A) $\tau = -m$ and $V_0 = b$
   • B) $\tau = \frac{1}{m}$ and $V_0 = \ln b$
   • C) $\tau = -\frac{1}{m}$ and $V_0 = b$
   • D) $\tau = -\frac{1}{m}$ and $V_0 = e^b$
7. 7.

   In a laboratory experiment to determine the focal length $f$ of a thin converging lens, a student measures the image distance $d_i$ for several different object distances $d_o$. The quantities are related by the thin lens equation, $\frac{1}{d_o} + \frac{1}{d_i} = \frac{1}{f}$. To linearize the non-linear data, the student plots $\frac{1}{d_i}$ on the vertical axis against $\frac{1}{d_o}$ on the horizontal axis and fits a straight line with slope $m$ and vertical intercept $b$. According to the theoretical model, what is the expected slope $m$ of this line, and how can the focal length $f$ be determined from the vertical intercept $b$?

   • A) The slope is $m = +1$, and the focal length is $f = \frac{1}{b}$.
   • B) The slope is $m = -1$, and the focal length is $f = b$.
   • C) The slope is $m = -1$, and the focal length is $f = \frac{1}{b}$.
   • D) The slope is $m = -\frac{1}{f}$, and the focal length is $f = -\frac{b}{m}$.
8. 8.

   To determine the mass $M$ of a central body, an astrophysics student collects orbital period $T$ and radius $r$ measurements for several moons in circular orbits governed by $T^2 = \left(\frac{4\pi^2}{GM}\right)r^3$, where $G$ is the gravitational constant. The student linearizes the data by plotting $T^2$ on the vertical axis against $r^3$ on the horizontal axis and determines the slope $S$ of the resulting best-fit line. Which expression correctly calculates $M$ from the slope, and what are the SI units of $S$?

   • A) $M = \frac{GS}{4\pi^2}$, with units of $\text{s}^2/\text{m}^3$
   • B) $M = \frac{4\pi^2 S}{G}$, with units of $\text{s}/\text{m}$
   • C) $M = \frac{G}{4\pi^2 S}$, with units of $\text{m}^3/\text{s}^2$
   • D) $M = \frac{4\pi^2}{GS}$, with units of $\text{s}^2/\text{m}^3$
9. 9.

   A student investigates the refraction of light traveling from air, with refractive index $n_1 = 1.00$, into an unknown liquid with refractive index $n_2$. According to Snell's law, $n_1 \sin\theta_1 = n_2 \sin\theta_2$, where $\theta_1$ is the angle of incidence and $\theta_2$ is the angle of refraction. To linearize the experimental data, the student plots $\sin\theta_1$ on the vertical axis against $\sin\theta_2$ on the horizontal axis and fits a straight line passing through the origin. What does the slope of this best-fit line represent, and what would a steeper slope indicate about the speed of light in the liquid?

   • A) The slope equals $1/n_2$, and a steeper slope indicates a lower speed of light in the liquid.
   • B) The slope equals $1/n_2$, and a steeper slope indicates a higher speed of light in the liquid.
   • C) The slope equals $n_2$, and a steeper slope indicates a higher speed of light in the liquid.
   • D) The slope equals $n_2$, and a steeper slope indicates a lower speed of light in the liquid.
10. 10.

    A student investigates how the current $I$ in a direct-current circuit depends on an adjustable external resistance $R$, using a battery with unknown electromotive force $\mathcal{E}$ and internal resistance $r$. The theoretical relationship between these variables is given by $I = \frac{\mathcal{E}}{R + r}$. To linearize the data, the student plots $\frac{1}{I}$ on the vertical axis against $R$ on the horizontal axis and fits a straight line with slope $S$ and vertical intercept $b$. Which of the following expressions correctly determines the electromotive force $\mathcal{E}$ and the internal resistance $r$ from this linear fit?

    • A) $\mathcal{E} = S$ and $r = b$
    • B) $\mathcal{E} = \frac{1}{S}$ and $r = \frac{b}{S}$
    • C) $\mathcal{E} = \frac{1}{S}$ and $r = b S$
    • D) $\mathcal{E} = \frac{1}{b}$ and $r = \frac{S}{b}$
11. 11.

    In a laboratory investigation, students measure the terminal speed $v$ of spheres with different radii $r$ falling through a viscous fluid to test a proposed model $v = C r^n$, where $C$ and $n$ are positive constants. To linearize the non-linear relationship, they construct a graph by plotting $\ln v$ on the vertical axis against $\ln r$ on the horizontal axis, which yields a straight line with slope $m$ and vertical intercept $b$. Which of the following correctly relates the model parameters $n$ and $C$ to the graphical parameters $m$ and $b$?

    • A) $n = e^m$ and $C = b$
    • B) $n = b$ and $C = e^m$
    • C) $n = m$ and $C = b$
    • D) $n = m$ and $C = e^b$
12. 12.

    A student investigates how the illuminance $E$ produced by a small light source varies with distance $r$, following the theoretical relationship $E = \frac{k}{r^2}$, where $k$ is a source constant. A raw plot of $E$ versus $r$ yields an inverse-square curve. Which variable should the student plot on the horizontal axis against $E$ on the vertical axis to linearize the data, and how can the constant $k$ be determined from the resulting line?

    • A) Plot $r^2$ on the horizontal axis; $k$ is equal to the slope of the line.
    • B) Plot $1/r^2$ on the horizontal axis; $k$ is equal to the slope of the line.
    • C) Plot $1/r$ on the horizontal axis; $k$ is equal to the square of the slope.
    • D) Plot $r$ on the horizontal axis; $k$ is equal to the vertical intercept of the line.
13. 13.

    In a laboratory investigation, students measure the speed $v$ of a cart at various positions $x$ along a track as it accelerates uniformly from an initial speed $v_0$ with constant acceleration $a$. According to kinematic theory, these variables satisfy the relation $v^2 = v_0^2 + 2ax$. To linearize their data, the students plot $v^2$ on the vertical axis versus $x$ on the horizontal axis and obtain a straight line. How should the students determine the values of $v_0$ and $a$ from the resulting best-fit line?

    • A) Taking the square root of the vertical intercept yields $v_0$, and dividing the slope by 2 yields $a$.
    • B) The vertical intercept directly equals $v_0$, and the slope directly equals $a$.
    • C) Taking the square root of the horizontal intercept yields $v_0$, and multiplying the slope by 2 yields $a$.
    • D) Dividing the vertical intercept by 2 yields $v_0$, and taking the square root of the slope yields $a$.
14. 14.

    A physics student investigates how the oscillation period T of a mass-spring system depends on the attached mass m, following the theoretical relationship $T = 2\pi\sqrt{m/k}$, where k is the spring constant. To linearize the experimental data so that the plotted points follow a straight line passing through the origin, the student places $T^2$ on the vertical axis. Which quantity should the student plot on the horizontal axis, and what theoretical expression corresponds to the slope of this best-fit line?

    • A) Plot $m$ on the horizontal axis; the slope equals $\frac{k}{4\pi^2}$.
    • B) Plot $m$ on the horizontal axis; the slope equals $\frac{4\pi^2}{k}$.
    • C) Plot $\sqrt{m}$ on the horizontal axis; the slope equals $\frac{4\pi^2}{k}$.
    • D) Plot $m^2$ on the horizontal axis; the slope equals $\frac{2\pi}{\sqrt{k}}$.