Simple Pendulum

NCERT Class 11, Chapter 14 gives the period of a simple pendulum executing small oscillations as $T=2\pi \sqrt{\frac{L}{g}}$, which the laboratory experiment rearranges to find $g=\frac{4{\pi }^{2}L}{T^2}$. Substituting the measured length $L=1.0$ m and period $T=2.0$ s gives $g=\frac{4{\pi }^2\times 1.0}{(2.0{)}^2}=\frac{4\times 9.87}{4}=9.87$ m/s${}^2$. This closely matches the accepted value of gravitational acceleration, confirming the method. The value 4.93 m/s${}^2$ is wrong because it drops the factor of 4 in the numerator, halving the numerator incorrectly. The value 19.7 m/s${}^2$ is wrong because it forgets to square the period in the denominator, leaving $T=2$ instead of $T^2=4$. The value 1.0 m/s${}^2$ is wrong because it merely reports the length numerically, ignoring the formula altogether. The derivation relies on the restoring torque for small angular displacements being proportional to the displacement, which produces simple harmonic motion with a period independent of both amplitude and bob mass. In practice the length is measured to the centre of the bob, and the period is found by timing twenty or more oscillations to reduce timing error, then dividing by the number of swings. A magnitude check confirms the result lands near the standard 9.8 m/s${}^2$, exactly what a one-metre pendulum with a two-second period should yield, validating the experimental setup.