Anand M. answered • 21d

The Keys to Understanding a Subject: Fundamentals and Problem Solving

Let's start by writing down the information which was given:

- t
_{0}= time for the oscillations - f
_{1}= 30 / t_{0}[oscillations / sec] - f
_{2}= 15 / t_{0}[oscillations / sec] = (1/2) f_{1} - L
_{2}- L_{1}= 3 [m]

and our unknowns are

- L
_{1}and L_{2}

It is useful to notice that we had to add a time, but the problem left it unspecified. The most likely thing we will end up doing is taking a ratio so the t_{0} factors from each frequency cancels out.

Since we are dealing with the motion of a pendulum of particular lengths, the relevant relationship is

- T = 2 π √(L/g)

Our unknowns are the lengths, so lets start by solving this for L,

- L = g (T / 2π)
^{2}

We were given oscillations for some amount of time, so to get the period of each pendulum, we use the relationship

- T = 1/f

The ratio of the lengths will simply the expression,

- L
_{1}/ L_{2}= ( T_{1}/ T_{2 })^{2}= ( f_{2}/ f_{1})^{2}= ( 1/2 )^{2}= 1/4

Knowing the ratio and the difference of the lengths is enough to calculate each value.