The motion of a pendulum can be described by the following differential equation
ID: 1910700 • Letter: T
Question
The motion of a pendulum can be described by the following differential equation [eq. (1)] when the angle of oscillation is small. Fig. 6 A pendulum. Show that a solution in the following form satisfies the above differential equation. x(t) = Asin(omega t + ) (2) Express the angular frequency in terms of quantities given in the differential equation (1). Describe the physical meaning of A in the solution (2). The initial condition is x(0) = A. Under this condition, determine the value of the initial phase .Explanation / Answer
let w= sqrt(g/L) x(t) = [(A+B) cos wt + i(A-B) sin wt ] this can be written as x(t) = C sin (wt + delta) where C = sqrt [ (A+B)^2 - (A-B)^2] delta = tan^-1 (A-B/A+B) a) x(t) = A sin (wt + delta ) where w = sqrt(g/L) b) w = sqrt(g/L) c) A is the amplitude of oscillation d) x(0) = A put t= 0 x(0) = A sin delta = A sin delta = 1 delta = pi/2