Dimensional Analysis: Deriving the Formula of any Physical Quantity

Transcript

Dimensional Analysis: Deriving the Formula of any Physical Quantity To derive a relationship using dimensional analysis: Identify the quantities on which the required quantity depends. Assume a product of powers of those quantities. Equate dimensions and compare the powers of each base quantity. EXAMPLE 1: FORCE Force depends on mass and acceleration. Assume: F = Kmˣaʸ K is a dimensionless constant. [F] = [MLT⁻²] [m] = [M] [a] = [LT⁻²] Therefore: [MLT⁻²] = [MˣLʸT⁻²ʸ] Comparing powers of M: x = 1 Comparing powers of L: y = 1 Comparing powers of T: −2y = −2 Hence: F = Kma Dimensional analysis gives the relationship but cannot determine K. EXAMPLE 2: CENTRIPETAL FORCE Centripetal force depends on mass m, speed v and radius r. Assume: F = Kmˣvʸrᶻ [MLT⁻²] = [Mˣ(LT⁻¹)ʸLᶻ] [MLT⁻²] = [MˣLʸ⁺ᶻT⁻ʸ] Comparing powers of M: x = 1 Comparing powers of T: −y = −2 y = 2 Comparing powers of L: y + z = 1 2 + z = 1 z = −1 Therefore: F = Kmv²/r Dimensional analysis cannot determine the dimensionless constant K.