A man walking briskly in rain with speed υ must slant his umbrella forward making an angle θ with the vertical. A student derives the following relation between θ and v : tan q = υ and checks that the relation has a correct limit: as υ→0, θ →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.
Given relation: tan θ = v
The dimension of a physical quantity is defined as the power to which the fundamental quantities are raised to express the physical quantity. It is represented as M a L b T c .
Velocity is represented in m/sec
Hence, dimension of R.H.S. is M 0 L 1 T − 1 .
Since, trigonometric function is considered to be dimensionless quantity therefore,
Dimension of L.H.S. is M 0 L 0 T 0 .
L .H .S . ≠ R .H .S . .Therefore, this relation is incorrect dimensionally.
Make R.H.S. dimensionless to make the above relation correct.
Divide by some velocity v ′ so as to cancel the dimension of velocity in the numerator.
Thus, relation can be written as,
tan θ = v v ′
Thus, the correct relation is tan θ = v v ′ .
Given relation:
The dimension of a physical quantity is defined as the power to which the fundamental quantities are raised to express the physical quantity. It is represented as
Velocity is represented in
Hence, dimension of R.H.S. is
Since, trigonometric function is considered to be dimensionless quantity therefore,
Dimension of L.H.S. is
Make R.H.S. dimensionless to make the above relation correct.
Divide by some velocity
Thus, relation can be written as,
Thus, the correct relation is
