Dimensional Analysis Problems

Dimensional Analysis - Problem 1

• The pressure per unit of length depends on the weight of the water through the pipe, the speed of the water, and the acceleration of gravity.
• These factors are related by a constant, k, which is determined experimentally.

Dimensional Analysis - Problem 2

• To ensure dimensional homogeneity, the dimensions of "Q" must be determined.
• The expression involves the following variables:
  • m = mass
  • R = radius
  • x = time
  • B = force
  • Z = velocity
• The dimensions of "Q" can be calculated by analyzing the dimensions of each variable.

Dimensional Analysis - Problem 3

• The equation for a physical phenomenon is given.
• The equation involves velocity (V), force (F), and acceleration (a).
• The dimensions of "z" need to be determined.

Dimensional Analysis - Problem 4

• The expression is: $\sum_{i=1}^{n}D_1ce= \frac{e^{-Pvt}}{D_0}$
• The variables involved are:
  • v = linear velocity
  • D0, D1 = Density
  • c, e = length
  • t = time
• The value of "P" needs to be chosen to ensure that the expression is dimensionally correct.

Dimensional Analysis - Problem 5

• The equation is: K2=F6PD2V−1K^2 = \frac{F}{6\sqrt{PD^2V^{-1}}}K2=6PD2V−1​F​
• The variables involved are:
  • F = Force
  • P = Pressure
  • D = Density
  • v = Velocity.
• The dimensional equation for "K" needs to be determined.

Dimensional Analysis - Problem 6

• The density of the magnetic flux "B" is given by: B=μI2πrB = \frac{\mu I}{2\pi r}B=2πrμI​
• The variables involved are:
  • B = Magnetic flux density
  • I = Current
  • μ = Magnetic permeability
  • r = Radial distance
• The unit of the magnetic permeability "μ" needs to be determined in the SI unit system.

Dimensional Analysis - Problem 7

• The equation is: FV=9,8∗P∗5∗m∗sen(37°)X\frac{F}{V} = \frac{9,8 * P * \sqrt{5} * m * sen(37°) }{X}VF​=X9,8∗P∗5​∗m∗sen(37°)​
• The variables involved are:
  • F = Force
  • V = Volume
  • P = Power
  • m = Length
  • X = Unknown variable
• The dimensional equation for "X" needs to be determined.

Dimensional Analysis - Problem 8

• The critical velocity "v" of a liquid flowing through a pipe depends on:
  • viscosity "η"
  • density "ρ"
  • diameter "D"
  • a dimensionless constant "R"
• The dimensions of "v" need to be expressed in terms of the dimensions of "η", "ρ", "D", and "R".

Dimensional Analysis - Problem 9

• A new unit system is defined where velocity, mass, and force are the fundamental magnitudes.
• The new system uses A for velocity, B for mass, and C for force.
• The dimensional equation of "E" needs to be determined in this new system, knowing that E = pressure x density.

Dimensional Analysis - Problem 10

• The electric potential energy (UE) is given by: UE=kQ2dU_E = \frac {kQ^2}{d}UE​=dkQ2​
• The variables are:
  • Q = Electric charge
  • d = Distance.
• The dimensions of "k" needs to be determined.