# Question #7c91f

May 16, 2015

My guess would be ${T}^{2} = \frac{4 {\pi}^{2}}{g} \cdot \left(\frac{{L}_{1} + {L}_{2}}{2}\right)$
By obstructed I am assuming the pendulum bob is on a string that hits a peg or other obstruction at the halfway point of its swing (see diagram)

I'm not 100% sure but believe the period of this type of pendulum will be the average of the period on the unobstructed side ${\text{Length}}_{1}$ and the obstructed side ${\text{Length}}_{2}$

The relationship between period and length is given as

So for ${T}^{2}$ the relationship would be

${T}^{2} = \frac{4 {\pi}^{2} L}{g}$

The mean value of the two periods (and the period of the obstructed pendulum) would be equal to

$\frac{\frac{4 {\pi}^{2} {L}_{1}}{g} + \frac{4 {\pi}^{2} {L}_{2}}{g}}{2} \mathmr{and} \frac{4 {\pi}^{2}}{g} \cdot \frac{\left({L}_{1} + {L}_{2}\right)}{2}$