Linearizing data Knowledge-Based Download PDF

Questions

1. 1.

   A measured physical quantity $y$ depends on time $t$ according to the exponential equation $y = A e^{k t}$, where $A$ and $k$ are positive constants. Which plot will produce a straight line with a slope equal to $k$ and a vertical intercept equal to $\ln(A)$?

   • A) Plotting $\ln(y)$ versus $\ln(t)$
   • B) Plotting $y$ versus $e^t$
   • C) Plotting $\ln(y)$ versus $t$
   • D) Plotting $1/y$ versus $t$
2. 2.

   A student investigates an inverse relationship between two measured physical quantities modeled by the equation $y = \frac{k}{x}$, where $k$ is a constant. If the student plots $y$ on the vertical axis, which quantity should be plotted on the horizontal axis to obtain a straight-line graph passing through the origin?

   • A) $x$
   • B) $x^2$
   • C) $\frac{1}{x}$
   • D) $\sqrt{x}$
3. 3.

   When analyzing experimental data, a student plots $\log(y)$ on the vertical axis versus $\log(x)$ on the horizontal axis and obtains a straight line. Which type of functional relationship between $x$ and $y$ does this result demonstrate?

   • A) An exponential relationship, where $y = A e^{k x}$
   • B) A logarithmic relationship, where $y = A \ln(x) + B$
   • C) A linear relationship, where $y = m x + b$
   • D) A power-law relationship, where $y = k x^n$
4. 4.

   A student measures the distance $d$ traveled by an object accelerating from rest with constant acceleration $a$ over time $t$, described by $d = \frac{1}{2}a t^2$. If the data are linearized by plotting $d$ on the vertical axis versus $t^2$ on the horizontal axis, what does the slope of the resulting best-fit line represent?

   • A) $\frac{1}{2}a$
   • B) $a$
   • C) $2a$
   • D) $\frac{1}{4}a$
5. 5.

   An experiment investigates the relationship between the period $T$ and length $L$ of a simple pendulum, described by $T = 2\pi \sqrt{\frac{L}{g}}$. To linearize the data into a straight-line graph passing through the origin, which quantity should be plotted on the vertical axis against $L$ on the horizontal axis?

   • A) $T$
   • B) $\sqrt{T}$
   • C) $\frac{1}{T}$
   • D) $T^2$