Consider the physical quantities s, v, a, and t with dimensions [s] = L, [v] =LT, [a] =LT, [t]=T.
Determine whether each of the following equations is dimensionally consistent:
1) svt+0.5at
2) svt+0.5at
3) vsin (at/s)
The Correct Answer and Explanation is :
To determine whether the equations are dimensionally consistent, we need to ensure that every term in each equation has the same dimensions.
Equation 1: ( svt + 0.5at )
- ( s ): L (length)
- ( v ): L/T (length per time)
- ( t ): T (time)
- ( a ): L/T² (length per time squared)
Checking the dimensions of each term:
- ( svt ): L × (L/T) × T = ( L^2 )
- ( 0.5at ): 0.5 × (L/T²) × T = ( L/T )
Here, the first term has the dimension ( L^2 ) and the second term has ( L/T ), which are not the same. Therefore, the equation ( svt + 0.5at ) is dimensionally inconsistent.
Equation 2: ( svt + 0.5at )
This equation is the same as Equation 1 and therefore also dimensionally inconsistent.
Equation 3: ( v \sin(at/s) )
- ( v ): L/T (length per time)
- ( a ): L/T² (length per time squared)
- ( s ): L (length)
- ( t ): T (time)
Checking the dimensions of ( \sin(at/s) ):
- ( at/s ): (L/T²) × T / L = 1/T² × T / L = 1/T, which is dimensionless since ( \sin ) function requires a dimensionless argument.
The term ( v \sin(at/s) ) itself:
- ( v \sin(at/s) ): (L/T) × dimensionless = L/T
Since the term ( v \sin(at/s) ) has consistent dimensions of L/T, the equation is dimensionally consistent.
Conclusion
Equations 1 and 2 are dimensionally inconsistent because the terms within each equation do not share the same dimensions. Equation 3 is dimensionally consistent as the function ( \sin ) correctly takes a dimensionless input, and the remaining terms ( v ) maintain consistent dimensions throughout.
Generating an Image
Let’s create an image that visualizes this concept of dimensional analysis, featuring symbolic representations of each equation and their dimensional evaluations to visually indicate consistency and inconsistency.
Here is the image prompt for visualization:
- A diagram showing three equations. Each equation is paired with symbols representing their dimensional forms. Equation 1 and 2 will show mismatched dimensions, indicated by different shapes or colors, highlighting their inconsistency. Equation 3 will show harmonious symbols, indicating dimensional consistency. The background is educational, with a subtle grid, and each equation is clearly labeled for clarity.