(a) Using principle of homogeneity of dimensions, check the dimensional accuracy of the equation. h=2σd/rgcosθ
Question
(a) Using principle of homogeneity of dimensions, check the dimensional accuracy of the equation.
h = \frac{2\sigma d}{rg \cos \theta}
Solution
Sure, let's check the dimensional accuracy of the equation h = 2σd / rgcosθ step by step.
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Identify the dimensions of each variable in the equation.
- h (height) has the dimension of length [L].
- σ (surface tension) has the dimension of force per unit length, which is [M/T^2].
- d (diameter) has the dimension of length [L].
- r (density) has the dimension of mass per unit volume, which is [M/L^3].
- g (acceleration due to gravity) has the dimension of acceleration, which is [L/T^2].
- cosθ is a trigonometric function and has no dimensions.
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Substitute the dimensions into the equation.
- The left side of the equation (h) has the dimension [L].
- The right side of the equation (2σd / rgcosθ) has the dimension [M/T^2][L] / ([M/L^3][L/T^2]*1) = [L].
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Compare the dimensions on both sides of the equation.
- The dimensions on both sides of the equation are [L], which means the equation is dimensionally accurate.
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