Algebra for physics Knowledge-Based Practice

Questions

1. 1.

   The theoretical efficiency $\eta$ of a heat engine is given by the formula $\eta = 1 - \frac{T_C}{T_H}$, where $T_H$ is the temperature of the hot reservoir and $T_C$ is the temperature of the cold reservoir. Which of the following expressions correctly isolates the hot reservoir temperature $T_H$ in terms of $\eta$ and $T_C$?

   • A) $T_H = \frac{T_C}{1 + \eta}$
   • B) $T_H = T_C(1 - \eta)$
   • C) $T_H = \frac{T_C}{1 - \eta}$
   • D) $T_H = \frac{1 - T_C}{\eta}$
2. 2.

   An object travels with constant acceleration $a$ such that its displacement over a time interval $t$ satisfies $\Delta x = v_0 t + \frac{1}{2}at^2$. Which of the following expressions correctly isolates the initial velocity $v_0$?

   • A) $v_0 = \frac{\Delta x}{t} + \frac{1}{2}at$
   • B) $v_0 = \frac{\Delta x}{t} - \frac{1}{2}at$
   • C) $v_0 = \frac{2\Delta x - at^2}{t}$
   • D) $v_0 = \Delta x - \frac{1}{2}at$
3. 3.

   An object of mass $m$ and specific heat capacity $c$ absorbs an amount of heat $Q$, changing its temperature from an initial value $T_i$ to a final value $T_f$ according to $Q = mc(T_f - T_i)$. Which of the following expressions correctly isolates the final temperature $T_f$?

   • A) $T_f = \frac{Q - T_i}{mc}$
   • B) $T_f = T_i + \frac{Q}{mc}$
   • C) $T_f = \frac{Q}{mc} - T_i$
   • D) $T_f = \frac{mc}{Q} + T_i$
4. 4.

   The magnitude of the gravitational force between two masses $m_1$ and $m_2$ separated by a distance $r$ is given by $F = G\frac{m_1 m_2}{r^2}$. Which of the following expressions correctly isolates the distance $r$ in terms of the other variables?

   • A) $r = \frac{G m_1 m_2}{F^2}$
   • B) $r = \frac{F}{G m_1 m_2}$
   • C) $r = \sqrt{\frac{G m_1 m_2}{F}}$
   • D) $r = \sqrt{\frac{F}{G m_1 m_2}}$
5. 5.

   The kinetic energy of an object is given by $K = \frac{1}{2}mv^2$ and its linear momentum is given by $p = mv$. Which of the following expressions represents the kinetic energy $K$ in terms of $p$ and $m$?

   • A) $K = \frac{2p^2}{m}$
   • B) $K = \frac{p}{2m}$
   • C) $K = \frac{p^2}{2m}$
   • D) $K = \frac{pm^2}{2}$
6. 6.

   The thin lens equation relates focal length $f$, object distance $d_{\text{o}}$, and image distance $d_{\text{i}}$ through $\frac{1}{f} = \frac{1}{d_{\text{o}}} + \frac{1}{d_{\text{i}}}$. Which of the following expressions correctly isolates the image distance $d_{\text{i}}$ in terms of $f$ and $d_{\text{o}}$?

   • A) $d_{\text{i}} = f - d_{\text{o}}$
   • B) $d_{\text{i}} = \frac{f d_{\text{o}}}{d_{\text{o}} - f}$
   • C) $d_{\text{i}} = \frac{d_{\text{o}} - f}{f d_{\text{o}}}$
   • D) $d_{\text{i}} = \frac{f d_{\text{o}}}{d_{\text{o}} + f}$
7. 7.

   The relationship between final velocity $v$, initial velocity $v_0$, constant acceleration $a$, and displacement $\Delta x$ is given by $v^2 = v_0^2 + 2a\Delta x$. Which of the following expressions correctly isolates the acceleration $a$?

   • A) $a = \frac{v - v_0}{2\Delta x}$
   • B) $a = \frac{v^2 + v_0^2}{2\Delta x}$
   • C) $a = \frac{v^2 - v_0^2}{2\Delta x}$
   • D) $a = \frac{2\Delta x}{v^2 - v_0^2}$
8. 8.

   The period $T$ of a simple pendulum of length $L$ is given by the equation $T = 2\pi \sqrt{\frac{L}{g}}$, where $g$ is the acceleration due to gravity. Which of the following expressions correctly isolates $g$ in terms of the other variables?

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

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