Question

4. Suppose the period of some motion T can only depend on the following parameters: a length L, T(M, L, a). Find all combinations a mass M, and an acceleration a. We abbreviate this as T T(M, L, a) allowed by units? Don't forget that if you find some functionTallowed, then of course any multiple of Tallowed by any unitless constant(like π or 7) is also allowed! Extra Challenge! Same problem, except that the list of parameters is: a length L, two masses Mi and M2, and the acceleration g. What is different?

Answer 1

We can find out T = T(M, L, a) using dimension analysis approach T = KM^a L^b a^c [M^0 L^0 T^1] = [M^a][L]^b [LT^-2]^c equate dimension of both sides a = 0 b + c = 0 -2c = 1 a = 0, c = -1/2 and b = 1/2 T = K(L/a)^1/2 rightarrow T = K squareroot L/a K is any constant Above relation is independence of mass. Hence either take M or M_1 and M_2 result remains same