Dimensional Formula of w and k in the given equation


The equation given is:
y = A sin (wt - kx)

To obtain the dimensional formula of w and k, we need to analyze the dimensions of each term in the equation.

1. Dimensional analysis of y:
The variable "y" represents the displacement or amplitude of the wave. Its dimensional formula is [M^0 L^1 T^0], where M represents mass, L represents length, and T represents time.

2. Dimensional analysis of A:
The variable "A" represents the maximum amplitude of the wave. Its dimensional formula is the same as that of "y", [M^0 L^1 T^0].

3. Dimensional analysis of sin (wt - kx):
The trigonometric function "sin" is dimensionless, so it does not have any dimensions.

Now, let's analyze the dimensions of the terms inside the sine function:

3.1. Dimensional analysis of wt:
The product of angular frequency "w" and time "t" should have the same dimensions as that of an angle. The dimensions of an angle are [M^0 L^0 T^0]. Therefore, the dimensional formula of "wt" is [M^0 L^0 T^-1].

3.2. Dimensional analysis of kx:
The product of wave number "k" and distance "x" should have the same dimensions as that of an angle. The dimensions of an angle are [M^0 L^0 T^0]. Therefore, the dimensional formula of "kx" is [M^0 L^1 T^0].

4. Equating the dimensions:
In the given equation, the terms inside the sine function should have the same dimensions in order for the equation to be dimensionally consistent. Therefore, we can equate the dimensions of "wt" and "kx":

[M^0 L^0 T^-1] = [M^0 L^1 T^0]

5. Solving for the dimensions of w and k:
From the above equation, we can conclude that the dimensional formula of "w" is [T^-1] and the dimensional formula of "k" is [L^-1].

6. Final dimensional formulas:
The dimensional formula of "w" is [T^-1] and the dimensional formula of "k" is [L^-1] in the given equation y = A sin (wt - kx).

Therefore, the dimensional formulas of "w" and "k" are [T^-1] and [L^-1] respectively.