Dimensional analysis is a powerful tool in physics that allows us to analyze and derive relationships between physical quantities based on their dimensions. It helps in understanding the basic structure and scaling behavior of physical laws. However, dimensional analysis has its limitations and cannot provide the exact values or determine dimensionless constants of proportionality in physical formulas.
Here are some limitations of dimensional analysis:
Dimensional analysis does not provide information about dimensionless constants: While dimensional analysis can help identify the dependence of variables and provide information about the powers of variables in an equation, it cannot determine the numerical values of dimensionless constants. These constants need to be determined through experimental measurements or by theoretical calculations.
Dimensional analysis cannot capture non-linear relationships: Dimensional analysis assumes linear relationships between physical quantities. It is not able to capture non-linear relationships or interactions between variables.
Dimensional analysis does not account for dimensionless ratios: Dimensional analysis does not provide information about dimensionless ratios that may appear in physical formulas. These ratios may arise from underlying mathematical relationships or fundamental physical principles.
Regarding your question about deriving formulas with dimensionless constants, yes, it is possible to derive formulas where the constant of proportionality is not dimensionless. The gravitational constant, G, in Newton's law of gravitation is an example of such a constant. Its dimensions are [G] = [M^{ -1} L^3 T^{ -2}], where M represents mass, L represents length, and T represents time. The value of G cannot be derived from dimensional analysis alone, and its precise value is determined through experimental measurements.
In general, to determine the value of a dimensionful constant, additional experimental data, theoretical calculations, or physical principles are required. Dimensional analysis provides a useful starting point for understanding the basic structure of physical formulas but does not provide all the necessary information to determine the values of dimensionful constants.