A simple pendulum having bob of mass m and length of string l has time period of T. If the mass of the bob is doubled and the length of the string is halved, then the time period of this pendulum will be

The time period (T) of a simple pendulum is given by the formula T = 2π√(l/g), where 'l' is the length of the string and 'g' is the acceleration due to gravity [1]. Crucially, the time period is independent of the mass of the bob [1]. Therefore, doubling the mass of the bob has no effect on the period. However, the period is directly proportional to the square root of the length (T ∝ √l). If the length 'l' is halved (l' = l/2), the new time period T' becomes T' = 2π√(l/2g). By factoring out the change, T' = (1/√2) × 2π√(l/g), which simplifies to T' = T/√2. Thus, while the mass change is irrelevant, the reduction in length results in the time period becoming T divided by the square root of 2.

Sources

1. [1] Science-Class VII . NCERT(Revised ed 2025) > Chapter 8: Measurement of Time and Motion > THINK LIKE A SCIENTIST! > p. 110