Algebra for physics Transitional Reasoning Practice

Questions

1. 1.

   A mechanical engineer evaluates a damping mechanism where an ideal spring obeys Hooke's law, $F = k x$, in which $F$ is the restoring force magnitude, $k$ is the spring constant, and $x$ is the displacement from equilibrium. The elastic potential energy stored in the compressed spring is given by $U = \frac{1}{2} k x^2$. To calibrate a dynamic monitoring system without installing displacement sensors, the engineer must isolate the spring constant $k$ exclusively in terms of the restoring force $F$ and stored potential energy $U$. Which algebraic formula correctly expresses $k$?

   • A) $k = \frac{2U}{F^2}$
   • B) $k = \frac{F}{2U}$
   • C) $k = \frac{F^2}{2U}$
   • D) $k = \frac{2F^2}{U}$
2. 2.

   An optical engineer calibrates an automated digital camera system where a thin lens of focal length $f$ projects a focused image onto an electronic sensor. The arrangement satisfies the thin-lens equation $\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$, where $d_o$ represents the object distance and $d_i$ represents the image distance, with the image magnification magnitude defined as $m = \frac{d_i}{d_o}$. To program the lens actuator based on target magnification without measuring the internal sensor distance, the engineer must isolate $d_o$ strictly in terms of $f$ and $m$. Which algebraic formula correctly expresses $d_o$?

   • A) $d_o = \frac{m f}{m + 1}$
   • B) $d_o = \frac{(m - 1)f}{m}$
   • C) $d_o = (m + 1)f$
   • D) $d_o = \frac{(m + 1)f}{m}$
3. 3.

   A laboratory technician investigates a real battery with electromotive force $\mathcal{E}$ and internal resistance $r$ connected across an adjustable load resistor $R$. The terminal potential difference across the battery is described by $V = \mathcal{E} - I r$, while the current through the load resistor satisfies Ohm's law, $V = I R$. To calibrate the system using voltage measurements rather than direct current readings, the technician must isolate the load resistance $R$ strictly in terms of $\mathcal{E}$, $V$, and $r$. Which algebraic formula correctly expresses $R$?

   • A) $R = \frac{V r}{\mathcal{E} - V}$
   • B) $R = \frac{(\mathcal{E} - V) r}{V}$
   • C) $R = \frac{V r}{\mathcal{E} + V}$
   • D) $R = \frac{\mathcal{E} r}{V} + r$
4. 4.

   A particle physicist analyzes collision debris in a detector where a non-relativistic particle of mass $m$ moves at speed $v$. The particle's linear momentum is given by $p = mv$, while its kinetic energy is given by $K = \frac{1}{2}mv^2$. To determine the particle's identity without directly measuring its speed, the researcher needs an expression that isolates the mass $m$ entirely in terms of momentum $p$ and kinetic energy $K$. Which algebraic formula correctly expresses $m$?

   • A) $m = \frac{2K}{p^2}$
   • B) $m = \frac{p}{2K}$
   • C) $m = \frac{p^2}{K}$
   • D) $m = \frac{p^2}{2K}$
5. 5.

   An acoustic engineer designs an instrument string of length $L$ and mass $m$ subjected to a tension force $T_s$. The fundamental frequency of the vibrating string is given by $f = \frac{v}{2L}$, and the wave speed is governed by $v = \sqrt{\frac{T_s}{\mu}}$, where the linear mass density is $\mu = \frac{m}{L}$. To calibrate the manufacturing specifications without measuring wave speed directly, the engineer needs an equation that isolates the string mass $m$ exclusively in terms of $L$, $f$, and $T_s$. Which algebraic formula correctly expresses the mass $m$?

   • A) $m = \frac{4 L f^2}{T_s}$
   • B) $m = \frac{T_s}{4 L f^2}$
   • C) $m = \frac{T_s}{2 L f^2}$
   • D) $m = \frac{T_s L}{4 f^2}$
6. 6.

   A telemetry engineer analyzes the horizontal test run of an automated cart that starts from rest and moves with constant acceleration $a$. The cart covers a displacement $d$ according to $d = \frac{1}{2} a t^2$, while reaching a final kinetic energy $K = \frac{1}{2} m v^2$ with final velocity $v = a t$. To program a real-time tracking algorithm, the engineer must express the elapsed time $t$ exclusively in terms of the cart's mass $m$, displacement $d$, and final kinetic energy $K$. Which formula correctly expresses $t$?

   • A) $t = d \sqrt{\frac{2m}{K}}$
   • B) $t = 2d \sqrt{\frac{m}{K}}$
   • C) $t = d \sqrt{\frac{m}{2K}}$
   • D) $t = \sqrt{\frac{2md}{K}}$
7. 7.

   An engineer designs an artificial-gravity centrifuge where a habitat module travels in a circle of radius $r$ with constant speed $v$. The centripetal acceleration is given by the relation $a = \frac{v^2}{r}$, and the linear speed relates to the rotation period $T$ by $v = \frac{2\pi r}{T}$. To program the automated controller, the engineer needs an expression that gives the rotation period $T$ directly in terms of radius $r$ and target acceleration $a$. Which algebraic formula correctly expresses $T$?

   • A) $T = 2\pi \sqrt{\frac{a}{r}}$
   • B) $T = \frac{2\pi r}{\sqrt{a}}$
   • C) $T = 4\pi^2 \sqrt{\frac{r}{a}}$
   • D) $T = 2\pi \sqrt{\frac{r}{a}}$

Answers and step-by-step solutions appear after you finish the test.