Question

check the equation v=ka2ut, where a is the area of cross section of the pipe ,u is the speed of flow ,t is the time v is the volume of the water flowing through the pipe and k is a dimensionless constant.state whether the equation is correct or not​

Answer 1

Explanation:

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Answer 2

Given:  v = ka²ut, where a is the area of cross section of the pipe, u is the speed of flow, t is the time, v is the volume of the water flowing through the pipe and k is a dimensionless constant.

To find: Whether the equation is correct or not​.

Solution:

• During dimensional analysis, the three quantities that are considered are mass, length and time.
• One side of the equation contains volume and the formula of volume of the pipe is given as,

        $V = 2 \pi r^{2} l$

• Here, 2 and π are constants and hence, dimensionless. r is the radius and l is the length of the pipe.
• So, the dimensions of volume can be given as,

        $r^{2} * l = [L^{2} * L]$

                 $= [L^{3}]$

                 $= [M^{0} L^{3} T^{0}]$

• On the other side of the equation, k is dimensionless and hence, is not considered for dimensional analysis.
• The dimension of acceleration (a) is given as [LT⁻²]. But the acceleration is squared, so its dimensions will be [L²T⁻⁴].
• The dimensions of velocity are given as [LT⁻¹].
• The dimension of time is given as [T].
• So, the dimensions of the other side of the equation can be given as,

        $[L^{2}T^{-4} ] * [LT^{-1}] *[T] = [M^{0} L^{3} T^{-4}]$

• The dimensions of mass and length of both the sides of the equation match but the dimensions of time do not.

Therefore, the equation is not​ correct.