Question

A man walking briskly in rain with speed v must slant his umbrella forward making an angle q with the vertical. A student derives the following relationship between q and v; tan q = v and checks that the relation has a correct limit as: v ® 0, q ® 0 as expected. (We are assuming there is no strong wind and that the rain falls vertically for a stationary man). Do you think this relation can be correct? If not, guess the correct relation?

Answer 1

Hii there,

# Answer-
Incorrect, correct relation should be tanθ = ν/ν’

# Solution-
Given relation is
tanθ = ν
Dimension of R.H.S = [M^0L^1T^–1]
Dimension of L.H.S = [M^0L^0T^0]

Dimensions of two sides don't match, hence this relation is wrong on dimensional ground.

To make the given relation correct, the R.H.S should also be dimensionless. One way to achieve this is by dividing the R.H.S by the speed of rainfall ν’
Therefore, the relation reduces to
tanθ = ν/ν’


Hope you understand it well...

Answer 2

Answer:

Incorrect; on dimensional ground

The relation is tanθ=v  

Dimension of R.H.S = [M^ 0  L^ 1  T^ −1 ]

Dimension of L.H.S =[ M^0  L^0  T^ 0 ]

 ( The trigonometric function is considered to be a dimensionless quantity)  

Dimension of R.H.S is not equal to the dimension of L.H.S.

Hence, the given relation is not correct dimensionally.

To make the given relation correct, the R.H.S should also be dimensionless. One way to achieve this is by dividing the R.H.S by the speed of rainfall v ′

 

Therefore, the relation reduces to

tanθ=v/v ′

 

This relation is dimensionally correct.