Unit 1 Dimensional Analysis

Question 1

Units and Dimensions of Force

Answer 1

N {=} [ML/T²]

describes the acceleration of a given mass

Question 2

Dimensions of Pressure

Answer 2

Pa or N/m² {=} [M/LT²]

describes the force over a particular area

Question 3

Dimensions of Energy

Answer 3

J or N⋅m {=} [ML²/T²]

describes the force produced over a particular distance

Question 4

Dimensions for Power

Answer 4

W or J/s {=} [ML²/T³]

describes the energy produced over a particular time interval

Question 5

Dimensions for Voltage

Answer 5

V or W/A {=} [ML²/T³I]

describes the power produced over a particular current

Question 6

List the Fundamental Dimensions [and example unit].

Answer 6

M = Mass [kg]
L = Length [m]
T = Time [s]
θ = Temperature [K]
I = Electric Current [A]
J = Light Intensity [cd]
N = Amount of Substance [mol]

Question 7

Explain process of finding dimensions of undefined variable in an Equation:

Answer 7

• Start by converting the given units into familiar/fundamental units
• Perform algebra to separate the undefined variable:
• Substitue the values with fundamental units

Question 8

Explain Process of Determining Dimensional Consistency:

Answer 8

• Start by converting the given units into familiar/fundamental units
• Convert the fundamental units of each side into dimensions
• Check dimensions are equivalent

Question 9

Describe how to find the dimensions of variable ω in the equation:

E = ωΦ²/T

Where:
E [=] Calories per mole [cal/mol]
Φ [=] Amps per second [A/s]
T [=] Kelvin [K]

Answer 9

Start by converting the given units into familiar/fundamental units:

E [=] Calories per mole [cal/mol] → [kgm²/s²mol]

Perform algebra to separate the undefined variable:

E = ωΦ²/T → ω = TE/Φ²

Substitue the values with fundamental units:

ω = TE/Φ² → ω = K⋅kgm²s²/s²mol⋅A²

Convert the fundamental units into dimensions:

ω = K⋅kg⋅m²⋅s²/molA² → ω = θML²/NI²

Question 10

Describe how to find the dimensions of variable η in the equation:

T = ηρ⁻¹

Where:
T [=] Temperature [F°]
ρ [=] Molar Density [mol/ft³]

Answer 10

Start by converting the given units into familiar/fundamental units:

T [=] Temperature {=} F° → [K]
ρ [=] Molar Density {=} mol/ft³ → **[mol/m³]

Perform algebra to separate the undefined variable:

T = ηρ⁻¹ → η = Tρ

Substitue the values with fundamental units:

η = Tρ → η = K⋅mol/m³

Convert the fundamental units into dimensions:

η = K⋅mol/m³ → η = θN/L³

Question 11

Describe how to find the value of n in the equation:

σ = F⋅Lⁿ/3E⋅I

Where:
σ [=] Displacement [in]
E [=] Youngs Modulus [lbf/in²]
I [=] Moment of Inertia [in⁴]
F [=] Force [lbf]
L [=] Column Length [in]

Answer 11

Convert the given units into familiar/fundamental units:

E [=] Youngs Modulus [lbf/in²] → [N/in²]
F [=] Force [lbf] → [N]

Perform algebra to separate the variable attached to our undefined value:

σ = F⋅Lⁿ/3E⋅I → Lⁿ = σ⋅3E⋅I/F

Substitue the values with fundamental units and cancel repeat units:

Lⁿ = σ⋅3E⋅I/F → inⁿ = in⋅N⋅in⁴/N⋅in² → inⁿ = in³

Perform algebra to separate solve for n

inⁿ = in³ → n⋅ln(in) = 3⋅ln(in) → n = 3

Question 12

Explain the dimensions of P/X and S/Y for:

Force F is given as F = X Cos(Pt) + Y Sin (Qs). t and s are time and distance

Answer 12

Since we know that the Dimensions of F = [M¹L¹T⁻²] We must assume that X and Y (given that they’re being added together) are both the dimensions of F.

We also know that a Trig operator and the variables operating within it must become dimensionless so: Pt and Qs dimensions are both [M⁰L⁰T⁰].

Since we know t = time [=] [M⁰L⁰T¹] & s = distance [=] [M⁰L¹T⁰]
we can set up our two equations to find P and Q as follows:

Pt = [M⁰L⁰T⁰] → P[M⁰L⁰T¹] = [M⁰L⁰T⁰] → P = [M⁰L⁰T⁻¹]

Qs = [M⁰L⁰T⁰] → Q[M⁰L¹T⁰] = [M⁰L⁰T⁰] → Q = [M⁰L⁻¹T⁰]

Now we can set up P/X and Q/Y using our two new dimensions:

P/X → [M⁰L⁰T⁻¹]/[M¹L¹T⁻²] = [M⁻¹L⁻¹T¹]

Q/Y → [M⁰L⁻¹T⁰]/[M¹L¹T⁻²] = [M⁻¹L⁻²T²]