I have x\'\' + g*sin(x)/L = 0, where x = theta. Assume theta is small and linear
ID: 2979092 • Letter: I
Question
I have x'' + g*sin(x)/L = 0, where x = theta. Assume theta is small and linearize the equation with respect to theta using the first-order (linear) term in Taylor series expansion. Solve the resulting equation analytically with the initial conditions x = x(naught) and dx/dt = 0 at t = 0. What is the circular frequenecy omega? What is the period of oscillations, T? Using dimensional analysis, express omega in terms of other parameters (up to a constant dimensionless coefficient). How does the frequency change when the string length is tripled, 3L; when it is decreased to L/3?Explanation / Answer
for small x sin(x) = x
so x'' + g x/L = 0
x'' = - g x/L
this has solutions
x = A sin w t + B cos wt, where w = sqrt(g/L)
applying conditions
x0 = A sin 0 + B cos 0 means B = x0
x' = 0 means A = 0
x = x0 cos wt
T = 2 pi/w = 2 pi sqrt(L/g)
if tripled w goes down by a factor of sqrt(1/3)
if decreased by 1/3 goes up by a factor of sqrt(3)