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Questions and Answers
What does the variation of pressure per unit of length depend on?
The weight of the water through the pipe, the speed of the water, and the acceleration of gravity.
Determine the dimensions of 'Q' from the equation: $Sen\theta = \frac{PRx + QBz}{m}$.
The dimensions of 'Q' need to be established based on dimensional homogeneity.
What are the dimensions of 'z' in the equation: $V = \frac{3V^2aFy}{Sen(zay)} - xF$?
The dimensions of 'z' must be derived from the equation based on dimensional consistency.
What should be the value of 'P' for the expression: $\sum_{i=1}^{n}D_1ce= \frac{e^{-Pvt}}{D_0}$ to be dimensionally correct?
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Find the dimensional equation of 'K' from: $K^2 = \frac{F}{6\sqrt{PD^2V^{-1}}}$.
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What is the unit of magnetic permeability μ in the equation: $B = \frac{\mu I}{2\pi r}$?
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Find the dimensional equation of 'X' in the relationship: $\frac{F}{V} = \frac{9.8 * P * \sqrt{5} * m * sen(37°) }{X}$.
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What is the dependence of 'v' on viscosity 'η', density 'p', diameter 'D', and constant 'R'?
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Find the dimensional equation of 'E' in a unit system where velocity, mass, and force are fundamental.
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What is the value of [K] in the equation: $U_E = \frac {kQ^2}{d}$?
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Study Notes
Dimensional Analysis Problems
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Problem 1:
- The variation of pressure per unit length in a pipe depends on the weight of water, water speed, and gravitational acceleration.
- The goal is to determine a formula that represents this relationship using a constant 'k'.
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Problem 2:
- The given equation is: $Sen\theta = \frac{PRx + QBz}{m}$.
- The dimensions of 'm' are mass (M).
- The dimensions of 'R' are length (L).
- The dimensions of 'x' are time (T).
- The dimensions of 'B' are force (MLT⁻²).
- The dimensions of 'Z' are velocity (LT⁻¹).
- The objective is to find the dimensions of 'Q' to make the equation dimensionally homogeneous.
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Problem 3:
- The equation is: $V = \frac{3V^2aFy}{Sen(zay)} - xF$.
- The dimensions of 'V' are velocity (LT⁻¹).
- The dimensions of 'F' are force (MLT⁻²).
- The dimensions of 'a' are acceleration (LT⁻²).
- The goal is to determine the dimensions of 'z'.
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Problem 4:
- The equation is: $\sum_{i=1}^{n}D_1ce= \frac{e^{-Pvt}}{D_0}$.
- The dimensions of 'v' are velocity (LT⁻¹).
- The dimensions of 'D0' and 'D1' are density (ML⁻³).
- The dimensions of 'c' and 'e' are length (L).
- The dimensions of 't' are ML⁻¹T⁻¹.
- The objective is to find the value of 'P' that makes the equation dimensionally correct.
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Problem 5:
- The equation is: $K^2 = \frac{F}{6\sqrt{PD^2V^{-1}}}$.
- The dimensions of 'F' are force (MLT⁻²).
- The dimensions of 'P' are pressure (ML⁻¹T⁻²).
- The dimensions of 'D' are density (ML⁻³).
- The dimensions of 'V' are velocity (LT⁻¹).
- The objective is to find the dimensional equation of 'K'.
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Problem 6:
- The equation: $B = \frac{\mu I}{2\pi r}$.
- The objective is to find the SI unit of magnetic permeability 'μ' in terms of Henry (H).
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Problem 7:
- The equation is: $\frac{F}{V} = \frac{9,8 * P * \sqrt{5} * m * sen(37°) }{X}$.
- The dimensions of 'P' are power (ML²T⁻³).
- The dimensions of 'V' are volume (L³).
- The dimensions of 'F' are force (MLT⁻²).
- The dimensions of 'm' are length (L).
- The objective is to find the dimensional equation of 'X'.
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Problem 8:
- The critical velocity 'v' of a liquid flow depends on viscosity 'η', density 'ρ', pipe diameter 'D', and a dimensionless constant 'R'.
- The dimensions of 'η' are ML⁻¹T⁻¹.
- The objective is to determine the dependence of 'v' on 'η', 'ρ', 'D', and 'R'.
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Problem 9:
- A new system defines velocity (A), mass (B), and force (C) as fundamental magnitudes.
- Pressure is defined as 'E = pressure x density'.
- The objective is to find the dimensional equation of 'E' in this new system.
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Problem 10:
- The electric potential energy equation is: $U_E = \frac {kQ^2}{d}$.
- The objective is to find the dimensions of 'k' ([K]).
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Description
Test your understanding of dimensional analysis with these problems that explore the relationships between pressure, mass, velocity, and force. Each problem challenges you to derive dimensions and ensure equations are dimensionally homogeneous. Perfect for physics students looking to sharpen their analytical skills!
