A simple pendulum suspended vertically from a rigid support is moving simple harmonically with a period of 5 s between two extreme positions A and B as shown in the figure. The angular distance between A and B is 8 cm. If it takes t seconds for the pendulum to move from position B to position C, exactly midway O and C, then find `(6)/(5)t`.

Answer

1

Explanation

From the figure,
Amplitude ` = OA = OB = (1)/(2) AB = (8)/(2) = 4 cm `
`therefore OC = 2 cm`
Let the displacement of the pendulum by given by
`x = A sin(omegat + phi) " "...(i)`
where ` a = 4 cm and omega = 2pi//T = 2pi//5 rad//s `
Let us consider that at t = 0, pendulum is at B, i.e., x =a
So, `a = a sin (omega xx 0 + phi)`
`a = a sin phi`
`sin theta = 1` i.e., `phi = pi//2 `
Substituting this value in (i), we get
`x = a sin(omegat + (pi)/(2)) = a cos omegat = 4 cos omegat `
For x = 2 cm, we have
`2 = 4 cos omegat`
`rArr cos omegat = (1)/(2)`
`rArr omegat = (pi)/(3)`
`rArr t = (pi)/(3omegat) = (5pi)/(6pi) = (5)/(6)s`
`rArr (6)/(5)t = 1`