Dimensional analysis Transitional Reasoning Download PDF

Questions

1. 1.

   In an environmental fluid mechanics investigation, researchers study sediment transport along an open canal by analyzing the friction between flowing water and the channel bed. The boundary shear stress $\tau_{\text{b}}$ is defined as the frictional resistance force per unit surface area, while fluid density $\rho$ represents mass per unit volume. To characterize turbulence near the channel bed, hydraulic engineers define the shear velocity $u_*$ through the relationship $u_* = \sqrt{\frac{\tau_{\text{b}}}{\rho}}$. Given that force has fundamental dimensions $\text{M L T}^{-2}$, what are the fundamental dimensions of the shear velocity $u_*$?

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

   An environmental engineering team studies the dispersion of a chemical tracer in groundwater using Fick's first law, $J = -D \frac{\Delta C}{\Delta x}$, where $J$ is the mass flux and $D$ is the dispersion coefficient. In this transport model, mass flux $J$ represents mass transferred per unit area per unit time with fundamental dimensions $\text{M L}^{-2}\text{T}^{-1}$, concentration $C$ is mass per unit volume with dimensions $\text{M L}^{-3}$, and $x$ represents distance with dimension $\text{L}$. Based on the principles of dimensional analysis, what are the fundamental dimensions of the dispersion coefficient $D$?

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

   An acoustic testing laboratory monitors transverse vibrations traveling along an elastic beam, modeling the surface displacement with the function $y(x, t) = y_0 \cos(\alpha x - \beta t)$. In this model, $x$ represents position along the beam with fundamental dimension $\text{L}$, and $t$ represents elapsed time with fundamental dimension $\text{T}$. Because the argument of a trigonometric function must be dimensionless, both terms inside the cosine function are pure numbers. Based on dimensional analysis, what are the fundamental dimensions of the ratio $\frac{\beta}{\alpha}$?

   • A) $\text{L}^{-1}\text{T}$
   • B) $\text{L T}^{-2}$
   • C) $\text{L T}^{-1}$
   • D) $\text{L}^2\text{T}^{-1}$
4. 4.

   An orbital mechanics navigation team calculates the gravitational trajectory of a satellite passing between two research probes. The mutual gravitational attraction $F$ between two bodies of masses $m_1$ and $m_2$ separated by a distance $r$ is given by Newton's law of universal gravitation, $F = G \frac{m_1 m_2}{r^2}$. Given that force has fundamental dimensions $\text{M L T}^{-2}$, what are the fundamental dimensions of the universal gravitational constant $G$?

   • A) $\text{M L}^3\text{T}^{-2}$
   • B) $\text{M}^{-1}\text{L}^3\text{T}^{-2}$
   • C) $\text{M}^{-1}\text{L}^2\text{T}^{-2}$
   • D) $\text{M}^2\text{L}^{-3}\text{T}^2$
5. 5.

   In an aerosol physics laboratory, researchers study the deceleration of airborne micro-particles suspended in a stagnant gas. The drag force resisting particle motion depends on the gas's dynamic viscosity $\eta$ (having fundamental dimensions $\text{M L}^{-1}\text{T}^{-1}$) and the spherical particle's radius $r$. To characterize how rapidly a particle of mass $m$ decelerates due to fluid friction, the researchers define a characteristic stopping time $t_{\text{s}}$ with the fundamental dimension of time ($\text{T}$). Which of the following expressions correctly represents a quantity with the dimensions of time?

   • A) $\frac{\eta r^2}{m}$
   • B) $\frac{m \eta}{r}$
   • C) $\frac{\eta}{m r}$
   • D) $\frac{m}{\eta r}$
6. 6.

   In an architectural acoustics laboratory, engineers evaluate the sound insulation of a partition wall by measuring sound intensity $I$. Sound intensity is defined as the acoustic power $P$ transmitted perpendicularly per unit area $A$, expressed as $I = \frac{P}{A}$, where power represents the acoustic energy $E$ transferred per unit time $\Delta t$ ($P = \frac{E}{\Delta t}$). Given that energy $E$ has fundamental dimensions $\text{M L}^2\text{T}^{-2}$, what are the fundamental dimensions of sound intensity $I$?

   • A) $\text{M T}^{-3}$
   • B) $\text{M L}^2\text{T}^{-3}$
   • C) $\text{M L}^{-1}\text{T}^{-2}$
   • D) $\text{M T}^{-2}$
7. 7.

   A materials testing laboratory evaluates heat shields by measuring the rate of heat energy conducted per unit time, $P = \frac{Q}{\Delta t}$, across a protective panel of cross-sectional area $A$ and thickness $\Delta x$ under a steady temperature difference $\Delta T$. The conduction rate is governed by the relation $P = k A \frac{\Delta T}{\Delta x}$, where $k$ is the thermal conductivity of the material. Given that energy $Q$ has fundamental dimensions $\text{M L}^2\text{T}^{-2}$ and temperature has fundamental dimension $\Theta$, what are the fundamental dimensions of $k$?

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

   During high-velocity impact testing of a composite shield, an engineering team evaluates the material response using a diagnostic performance metric. They define an indicator parameter $\Pi = \frac{E}{F \Delta t}$, where $E$ represents the total absorbed energy, $F$ is the average contact force, and $\Delta t$ is the collision duration. Based on the principles of dimensional analysis, which of the following physical quantities has the exact same fundamental dimensions as $\Pi$?

   • A) Acceleration
   • B) Speed
   • C) Power
   • D) Frequency
9. 9.

   In a mechanical testing laboratory, an engineer analyzes the load-displacement graph obtained from compressing an automotive suspension spring. The vertical axis represents the applied compressive force $F$ with fundamental dimensions $\text{M L T}^{-2}$, and the horizontal axis represents the displacement $x$ with fundamental dimension $\text{L}$. To configure an automated data-acquisition system, the engineer determines the fundamental dimensions for both the slope of the graph and the area under the curve. Which pair of fundamental dimensions correctly represents the slope and the area under this graph, respectively?

   • A) Slope: $\text{M L}^2\text{T}^{-2}$; Area: $\text{M T}^{-2}$
   • B) Slope: $\text{M T}^{-2}$; Area: $\text{M L}^2\text{T}^{-2}$
   • C) Slope: $\text{M L}^{-1}\text{T}^{-2}$; Area: $\text{M L}^3\text{T}^{-2}$
   • D) Slope: $\text{M L T}^{-3}$; Area: $\text{M L T}^{-1}$
10. 10.

    An engineering team evaluates energy storage systems by monitoring the volumetric energy density, defined as the stored mechanical energy per unit volume. During data validation, the team verifies that all measured telemetry channels are dimensionally consistent before combining them into composite performance metrics. Based on the principles of dimensional analysis, which of the following physical quantities has the exact same fundamental dimensions as volumetric energy density?

    • A) Power delivered per unit area
    • B) Force exerted per unit length
    • C) Change in momentum per unit time
    • D) Force exerted per unit area
11. 11.

    An acoustics engineering team models the attenuation of ultrasound waves traveling through bone tissue using the relation $A(x) = A_0 e^{-\mu x \sqrt{k/\rho}}$, where $x$ is the penetration depth, $\rho$ is the mass density with dimensions $\text{M L}^{-3}$, and $k$ is the bulk modulus with dimensions $\text{M L}^{-1}\text{T}^{-2}$. Because the exponent of an exponential function must be dimensionless, which fundamental dimensions must the attenuation coefficient $\mu$ possess?

    • A) $\text{L}^{-1}\text{T}$
    • B) $\text{L}^{-2}\text{T}$
    • C) $\text{L}^2\text{T}^{-1}$
    • D) $\text{L}^{-3}\text{T}^2$
12. 12.

    A biomedical engineering team investigates fluid transport through a microvascular vessel by modeling the volumetric flow rate $Q$ (volume per unit time). They evaluate candidate relationships that link $Q$ to the vessel radius $r$, the fluid's dynamic viscosity $\eta$ (dimensions $\text{M L}^{-1}\text{T}^{-1}$), and the axial pressure gradient $\frac{\Delta P}{\Delta x}$ (pressure change per unit length, where pressure has dimensions $\text{M L}^{-1}\text{T}^{-2}$). Based on the principle of dimensional homogeneity, which proposed expression correctly describes the volumetric flow rate?

    • A) $Q \propto \frac{\eta}{\Delta P / \Delta x} r^3$
    • B) $Q \propto \frac{\Delta P / \Delta x}{\eta} r^2$
    • C) $Q \propto \frac{\Delta P / \Delta x}{\eta} r^4$
    • D) $Q \propto \left(\frac{\Delta P / \Delta x}{\eta}\right)^2 r^3$
13. 13.

    In an applied thermodynamics investigation of non-ideal gases, researchers use the van der Waals equation of state, $\left(P + \frac{a}{V^2}\right)(V - b) = nRT$, to describe real gas behavior. According to the principle of dimensional homogeneity, two physical quantities can only be added together if they possess identical fundamental dimensions, meaning $\frac{a}{V^2}$ must share the dimensions of pressure $P$. Given that pressure has dimensions of $\text{M L}^{-1}\text{T}^{-2}$ and volume $V$ has dimensions of $\text{L}^3$, what are the fundamental dimensions of the parameter $a$?

    • A) $\text{M L}^2\text{T}^{-2}$
    • B) $\text{M L}^5\text{T}^{-2}$
    • C) $\text{M L}^{-4}\text{T}^{-2}$
    • D) $\text{M L}^4\text{T}^{-2}$
14. 14.

    In a study of fluid transport through a porous filter, an engineer models the pressure drop per unit length using the relationship $\frac{\Delta P}{L} = \alpha v + \beta v^2$, where $\Delta P$ represents pressure, $L$ represents length, and $v$ is the fluid flow speed. For the equation to be physically valid, every additive term must share the same dimensions as the left side of the equation. Given that pressure has dimensions of $\text{M L}^{-1}\text{T}^{-2}$, what are the fundamental dimensions of the coefficient $\beta$?

    • A) $\text{M L}^{-2}\text{T}^{-1}$
    • B) $\text{M L}^{-3}$
    • C) $\text{M L}^{-2}\text{T}^{-2}$
    • D) $\text{M L}^{-4}$
15. 15.

    A geophysicist modeling the propagation of seismic shear waves through rock layers proposes that wave speed $v$ depends solely on the shear modulus $G$ (having the dimensions of pressure, $\text{M L}^{-1}\text{T}^{-2}$), mass density $\rho$ (dimensions $\text{M L}^{-3}$), and a dimensionless constant $k$, expressed as $v = k G^a \rho^b$. To ensure the relationship is dimensionally homogeneous, which pair of exponents must satisfy this equation?

    • A) $a = 1$ and $b = -1$
    • B) $a = 1/2$ and $b = 1/2$
    • C) $a = -1/2$ and $b = 1/2$
    • D) $a = 1/2$ and $b = -1/2$
16. 16.

    An aerospace engineering team models the aerodynamic resistance force $F$ exerted on a vehicle moving through air as $F = C \rho^x v^y A^z$, where $C$ is a dimensionless coefficient, $\rho$ is the fluid density, $v$ is the relative speed, and $A$ is the frontal cross-sectional area. To ensure the mathematical model is dimensionally consistent, the dimensions of both sides must match in terms of mass ($\text{M}$), length ($\text{L}$), and time ($\text{T}$). Which set of exponents satisfies dimensional homogeneity for this relationship?

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