(a) Using principle of homogeneity of dimensions, check the dimensional accuracy of the equation. h=2σd​/rgcosθ

Sure, let's check the dimensional accuracy of the equation h = 2σd / rgcosθ step by step.

1. 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.

2. 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].

3. 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.

Sure, let's check the dimensional accuracy of the equation h = 2σd/(rgcosθ) using the principle of homogeneity of dimensions.

The principle of homogeneity of dimensions states that the dimensions on the left-hand side of an equation must be equal to the dimensions on the right-hand side.

Let's break down the dimensions of each term in the equation:

1. h (height) has dimensions of length (L).

2. σ (surface tension) has dimensions of force per unit length, which can be written as mass times acceleration divided by length (MT^-2).

3. d (diameter) also has dimensions of length (L).

4. r (density) has dimensions of mass per unit volume, which can be written as mass divided by length cubed (M L^-3).

5. g (acceleration due to gravity) has dimensions of length divided by time squared (L T^-2).

6. cosθ is a trigonometric function and is dimensionless.

Now, let's look at the right-hand side of the equation:

The dimensions of 2σd/(rgcosθ) can be written as (MT^-2 * L) / (M L^-3 * L T^-2) = L, which is the dimension of length.

Since the dimensions on both sides of the equation are the same (L = L), the equation is dimensionally accurate.