Question #78220
The velocity of elastic waves in a solid is found to depend on the density a, and Young's Modulus Y of the material of the solid. Given that Y=ML^1T-2, find the form of variation by the method of Dimensions.
Expert's answer
dimY=ML^(-1) T^(-2)
dimv=LT^(-1).
dimρ=ML^(-3).
v=Y^a ρ^b.
LT^(-1)=(ML^(-1) T^(-2) )^a (ML^(-3) )^b
M:0=a+b→b=-a
L:1=-a+3a=2a
a=1/2,b=-1/2.
Thus, velocity of elastic waves in a solid is given by the formula:
v=√(Y/ρ)
dimv=LT^(-1).
dimρ=ML^(-3).
v=Y^a ρ^b.
LT^(-1)=(ML^(-1) T^(-2) )^a (ML^(-3) )^b
M:0=a+b→b=-a
L:1=-a+3a=2a
a=1/2,b=-1/2.
Thus, velocity of elastic waves in a solid is given by the formula:
v=√(Y/ρ)
