Dimensional analysis Transitional Reasoning Practice

Questions

1. 1.

   A marine engineer evaluates structural materials for a deep-sea submersible hull subjected to high hydrostatic pressures. The bulk modulus $K$ of an alloy quantifies its resistance to uniform compression and is defined by $K = -V \frac{\Delta P}{\Delta V}$, where $V$ is volume, $\Delta V$ is the change in volume, and $\Delta P$ is the applied hydrostatic pressure. To quantify how easily the hull material deforms under pressure, the engineer calculates the compressibility $\beta = \frac{1}{K}$. Given that pressure has dimensions $[\text{M}][\text{L}]^{-1}[\text{T}]^{-2}$, what is the dimensional formula of the compressibility $\beta$?

   • A) $[\text{M}][\text{L}]^{-1}[\text{T}]^{-2}$
   • B) $[\text{M}][\text{L}]^2[\text{T}]^{-2}$
   • C) $[\text{M}]^{-1}[\text{L}][\text{T}]^2$
   • D) $[\text{M}]^{-1}[\text{L}]^{-1}[\text{T}]^2$
2. 2.

   To evaluate the radiative heat loss of an experimental satellite in deep space, aerospace engineers apply the Stefan–Boltzmann law, $I = \sigma T^4$, where $I$ is the radiant flux density and $T$ is the absolute temperature with dimension $[\Theta]$. Radiant flux density $I$ is defined as the radiant energy emitted per unit surface area per unit time, where energy has dimensions $[\text{M}][\text{L}]^2[\text{T}]^{-2}$. Based on the principle of dimensional homogeneity, what is the dimensional formula of the Stefan–Boltzmann constant $\sigma$?

   • A) $[\text{M}][\text{T}]^{-3}[\Theta]^{-4}$
   • B) $[\text{M}][\text{L}]^2[\text{T}]^{-3}[\Theta]^{-4}$
   • C) $[\text{M}][\text{T}]^{-2}[\Theta]^{-4}$
   • D) $[\text{M}][\text{L}]^{-2}[\text{T}]^{-3}[\Theta]^{-1}$
3. 3.

   An aerospace engineer evaluates thermal barrier coatings designed to protect spacecraft during atmospheric reentry. The rate of conductive heat transfer, $\dot{Q}$, across a coating layer of thickness $\Delta x$ and surface area $A$ is governed by Fourier's law, expressed as $\dot{Q} = k A \frac{\Delta T}{\Delta x}$. Here, $\dot{Q}$ represents thermal power transferred with dimensions $[\mathrm{M}][\mathrm{L}]^2[\mathrm{T}]^{-3}$, $A$ has dimensions $[\mathrm{L}]^2$, $\Delta x$ has dimension $[\mathrm{L}]$, and $\Delta T$ is the temperature difference with dimension $[\Theta]$. Based on the principle of dimensional homogeneity, what is the dimensional formula of the thermal conductivity $k$?

   • A) $[\mathrm{M}][\mathrm{L}]^2[\mathrm{T}]^{-3}[\Theta]^{-1}$
   • B) $[\mathrm{M}][\mathrm{L}]^{-1}[\mathrm{T}]^{-2}[\Theta]^{-1}$
   • C) $[\mathrm{M}][\mathrm{L}][\mathrm{T}]^{-2}[\Theta]^{-1}$
   • D) $[\mathrm{M}][\mathrm{L}][\mathrm{T}]^{-3}[\Theta]^{-1}$
4. 4.

   In biomedical inhaler design, engineers evaluate the breakup of liquid droplets into an inhalable mist by determining the Weber number, a dimensionless quantity comparing fluid inertia to surface tension. The droplet behavior depends on fluid density $\rho$ (dimensions $[\text{M}][\text{L}]^{-3}$), droplet velocity $v$ (dimensions $[\text{L}][\text{T}]^{-1}$), characteristic diameter $d$ (dimension $[\text{L}]$), and surface tension $\gamma$ (dimensions $[\text{M}][\text{T}]^{-2}$). Based on the principles of dimensional analysis, which of the following combinations yields a completely dimensionless group?

   • A) $\frac{\rho v d}{\gamma}$
   • B) $\frac{\rho v^2}{\gamma d}$
   • C) $\frac{\rho^2 v d}{\gamma}$
   • D) $\frac{\rho v^2 d}{\gamma}$
5. 5.

   In aerodynamic modeling, the turbulent kinetic energy dissipation rate per unit mass, $\varepsilon$, represents the rate at which fluid kinetic energy is converted into internal thermal energy per unit mass. Given that mechanical energy has dimensions $[\text{M}][\text{L}]^2[\text{T}]^{-2}$, the quantity $\varepsilon$ possesses the dimensional formula $[\text{L}]^2[\text{T}]^{-3}$. Assuming that $\varepsilon$ depends exclusively on the characteristic eddy fluctuation velocity $u$ (with dimensions $[\text{L}][\text{T}]^{-1}$) and the eddy scale length $l$ (with dimension $[\text{L}]$), which of the following expressions is dimensionally consistent, where $C$ is a dimensionless coefficient?

   • A) $\varepsilon = C \frac{u^2}{l}$
   • B) $\varepsilon = C \frac{u}{l^2}$
   • C) $\varepsilon = C \frac{u^2}{l^2}$
   • D) $\varepsilon = C \frac{u^3}{l}$
6. 6.

   A materials scientist evaluates the vibration-damping behavior of a viscoelastic polymer used in precision aerospace mounts. The internal stress relaxation in such polymers is governed by their dynamic viscosity $\eta$, with dimensions $[\text{M}][\text{L}]^{-1}[\text{T}]^{-1}$, and their elastic shear modulus $G$, which has dimensions of pressure, $[\text{M}][\text{L}]^{-1}[\text{T}]^{-2}$. To predict how rapidly mechanical stresses dissipate, the scientist defines a characteristic relaxation time $\tau$ with dimension $[\text{T}]$. Based on dimensional analysis, which of the following expressions correctly relates $\tau$ to $\eta$ and $G$?

   • A) $\tau = \sqrt{\frac{G}{\eta}}$
   • B) $\tau = G \eta$
   • C) $\tau = \frac{G}{\eta}$
   • D) $\tau = \frac{\eta}{G}$
7. 7.

   An environmental engineer investigates groundwater seepage through a packed soil column using Darcy's law, written as $Q = K A \frac{\Delta h}{L}$. Here, $Q$ is the volumetric flow rate (volume of fluid passing through the column per unit time), $A$ is the cross-sectional area of the column, $\Delta h$ is the hydraulic head difference (a length measurement), and $L$ is the column length. Based on the principle of dimensional homogeneity, what is the dimensional formula of the hydraulic conductivity $K$?

   • A) $[\text{L}][\text{T}]^{-1}$
   • B) $[\text{L}]^2$
   • C) $[\text{L}]^3[\text{T}]^{-1}$
   • D) $[\text{L}]^2[\text{T}]^{-1}$
8. 8.

   To model the passive transport of respiratory gases across a biological membrane, a biophysicist applies Fick's first law of diffusion, expressed as $J = -D \frac{\Delta C}{\Delta x}$. Here, the diffusion flux $J$ represents the number of molecules passing through a unit area per unit time, $\Delta x$ is the membrane thickness, and $\Delta C$ is the difference in molecular concentration (number of molecules per unit volume). Since the number of molecules is a dimensionless quantity, which of the following represents the correct dimensional formula for the diffusion coefficient $D$?

   • A) $[\text{L}]^2[\text{T}]^{-1}$
   • B) $[\text{L}][\text{T}]^{-1}$
   • C) $[\text{L}]^2[\text{T}]^{-2}$
   • D) $[\text{L}]^{-1}[\text{T}]^{-1}$
9. 9.

   To monitor vibrations in precision machinery, an engineer models the horizontal displacement $x(t)$ of an oscillating sensor component using the equation $x(t) = A e^{-\lambda t} \cos(\omega t + \phi)$, where $x$ represents position with dimension $[\text{L}]$, $t$ is time with dimension $[\text{T}]$, and $\phi$ is a constant phase angle. According to the principles of dimensional analysis, the overall equation must be dimensionally homogeneous and arguments of transcendental functions such as exponential and trigonometric functions must be dimensionless. Under these physical requirements, what are the dimensional formulas of the amplitude $A$, the decay constant $\lambda$, and the angular frequency $\omega$?

   • A) $[A] = [\text{L}][\text{T}]^{-1}$, $[\lambda] = [\text{T}]$, and $[\omega] = [\text{T}]^{-1}$
   • B) $[A] = [\text{L}]$, $[\lambda] = [\text{T}]^{-1}$, and $[\omega] = [\text{T}]^{-1}$
   • C) $[A] = [\text{L}]$, $[\lambda] = [\text{T}]$, and $[\omega] = [\text{T}]$
   • D) $[A] = [\text{L}][\text{T}]^{-2}$, $[\lambda] = [\text{T}]^{-1}$, and $[\omega] = 1$
10. 10.

    A biomedical engineering team designs a microfluidic device to separate circulating cells by sedimentation. To model the terminal drift speed $v$ of a spherical cell, researchers propose several equations relating cell radius $r$, fluid dynamic viscosity $\eta$, gravitational acceleration $g$, and density difference $\Delta \rho$. Given that speed has dimensions $[\text{L}][\text{T}]^{-1}$, acceleration has dimensions $[\text{L}][\text{T}]^{-2}$, density difference has dimensions $[\text{M}][\text{L}]^{-3}$, and dynamic viscosity has dimensions $[\text{M}][\text{L}]^{-1}[\text{T}]^{-1}$, which of the following proposed expressions is dimensionally consistent, where $C$ is a dimensionless constant?

    • A) $v = C \frac{\Delta \rho \cdot g \cdot r}{\eta}$
    • B) $v = C \frac{\Delta \rho \cdot g \cdot r^3}{\eta}$
    • C) $v = C \frac{\eta \cdot g}{\Delta \rho \cdot r^2}$
    • D) $v = C \frac{\Delta \rho \cdot g \cdot r^2}{\eta}$
11. 11.

    To account for intermolecular forces and finite molecular size, physicists model a real gas using the van der Waals equation $\left(P + \frac{a}{V^2}\right)(V - b) = RT$, where $P$ is pressure and $V$ is volume. According to the principle of dimensional homogeneity, terms added to or subtracted from each other must share identical dimensions. Given that pressure has dimensions $[\text{M}][\text{L}]^{-1}[\text{T}]^{-2}$ and volume has dimension $[\text{L}]^3$, what is the dimensional formula of the ratio $\frac{a}{b}$?

    • A) $[\text{M}][\text{L}]^{-1}[\text{T}]^{-2}$
    • B) $[\text{M}][\text{L}][\text{T}]^{-2}$
    • C) $[\text{M}][\text{L}]^5[\text{T}]^{-2}$
    • D) $[\text{M}][\text{L}]^2[\text{T}]^{-2}$
12. 12.

    In modern theoretical physics, fundamental constants are combined to construct characteristic natural units of measurement. A researcher investigates the dimensional properties of the expression $\sqrt{\frac{h c}{G}}$, where $h$ is Planck's constant with dimensions $[\mathrm{M}][\mathrm{L}]^2[\mathrm{T}]^{-1}$, $c$ is the speed of light with dimensions $[\mathrm{L}][\mathrm{T}]^{-1}$, and $G$ is the gravitational constant with dimensions $[\mathrm{M}]^{-1}[\mathrm{L}]^3[\mathrm{T}]^{-2}$. Which fundamental physical quantity does this algebraic combination represent?

    • A) Length $[\mathrm{L}]$
    • B) Mass $[\mathrm{M}]$
    • C) Time $[\mathrm{T}]$
    • D) Electric current $[\mathrm{I}]$
13. 13.

    A materials testing laboratory evaluates the propagation of sound waves in newly synthesized composite blocks. To quantify how much acoustic wave energy crosses from one layer to the next, the researchers analyze the characteristic acoustic impedance, which is defined as the ratio of acoustic pressure to particle speed. Given that pressure is defined as force per unit area and particle speed is length per unit time, which of the following represents the correct dimensional formula for acoustic impedance?

    • A) $[\mathrm{M}][\mathrm{L}]^{-1}[\mathrm{T}]^{-1}$
    • B) $[\mathrm{M}][\mathrm{L}]^{-3}[\mathrm{T}]^{-1}$
    • C) $[\mathrm{M}][\mathrm{L}]^{-2}[\mathrm{T}]^{-1}$
    • D) $[\mathrm{M}][\mathrm{L}]^0[\mathrm{T}]^{-2}$
14. 14.

    An oceanographic research station models the total wave energy per unit surface area, $E_A$, using the fluid density $\rho$ (dimensions $[\mathrm{M}][\mathrm{L}]^{-3}$), gravitational acceleration $g$ (dimensions $[\mathrm{L}][\mathrm{T}]^{-2}$), and wave height $H$ (dimension $[\mathrm{L}]$). Knowing that energy has dimensions $[\mathrm{M}][\mathrm{L}]^2[\mathrm{T}]^{-2}$, which of the following expressions is dimensionally consistent for $E_A$, where $C$ is a dimensionless constant?

    • A) $E_A = C \rho g H^2$
    • B) $E_A = C \rho g^2 H$
    • C) $E_A = C \rho^2 g H$
    • D) $E_A = C \rho g H$
15. 15.

    In an aerodynamics laboratory, researchers model the resistive force acting on a suspended particle using the empirical equation $F = 6\pi \eta r v + \beta t$, where $F$ is force, $r$ is particle radius, $v$ is speed, and $t$ is elapsed time. To ensure dimensional homogeneity, each term on the right-hand side must have the dimensions of force, given by $[\text{M}][\text{L}][\text{T}]^{-2}$. Knowing that radius has dimension $[\text{L}]$ and speed has dimensions $[\text{L}][\text{T}]^{-1}$, what are the dimensional formulas of the dynamic viscosity $\eta$ and the coefficient $\beta$?

    • A) $[\eta] = [\text{M}][\text{L}]^{-2}[\text{T}]^{-1}$ and $[\beta] = [\text{M}][\text{L}][\text{T}]^{-1}$
    • B) $[\eta] = [\text{M}][\text{L}][\text{T}]^{-1}$ and $[\beta] = [\text{M}][\text{L}][\text{T}]^{-2}$
    • C) $[\eta] = [\text{M}][\text{L}]^{-1}[\text{T}]^{-1}$ and $[\beta] = [\text{M}][\text{L}][\text{T}]^{-3}$
    • D) $[\eta] = [\text{M}][\text{L}]^{-1}[\text{T}]^{-2}$ and $[\beta] = [\text{M}][\text{L}][\text{T}]^{-3}$
16. 16.

    An engineering student models the power $P$ generated by a wind turbine as $P = C \rho^x R^y v^z$, where $C$ is a dimensionless constant, $\rho$ is air density, $R$ is blade radius, and $v$ is wind velocity. In SI base dimensions, power is expressed as $[\text{M}][\text{L}]^2[\text{T}]^{-3}$, density as $[\text{M}][\text{L}]^{-3}$, and velocity as $[\text{L}][\text{T}]^{-1}$. To ensure dimensional consistency across both sides of the equation, which set of exponents must the student determine?

    • A) $x = 1$, $y = 1$, $z = 2$
    • B) $x = 1$, $y = 2$, $z = 3$
    • C) $x = 2$, $y = 2$, $z = 1$
    • D) $x = 1$, $y = 3$, $z = 2$

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