The displacement y of a particle at position x and time t is given by the equation y = A (Bt - Cx) . If Force ( F ), Density ( ), and Time ( T ) are chosen as the fundamental physical quantities, what are the dimensions of the quantity AB C ?

Step-by-step solution

According to the principle of dimensional homogeneity, the arguments of trigonometric functions must be dimensionless. Thus, the quantities Bt and Cx are dimensionless. [B][T] = [M^0 L^0 T^0] [B] = [T⁻¹] [C][L] = [M^0 L^0 T^0] [C] = [L⁻¹] Since y represents displacement, [A] = [L] . The dimensional formula for the required quantity is: [ AB C ] = [L][T⁻¹] [L⁻¹] = [L^2 T⁻¹] Let [L^2 T⁻¹] = [F]^a [ ]^b [T]^c Substituting the standard dimensions for Force ( [MLT⁻²] ), Density ( [ML⁻³] ), and Time ( [T] ): [M^0 L^2 T⁻¹