Final answer:
To determine after how many vibrations two pendulums of lengths 1.44 meters and 1 meter will swing together again, calculate their periods using the formula T = 2π√(L/g), find the ratio of their periods, and identify the least common multiple (LCM) of these ratios.
Step-by-step explanation:
The question asks after how many vibrations two simple pendulums of different lengths will again start swinging together. To solve this, we need to find the periods of the pendulums and then determine how many cycles each will complete before their motions align.
Using the formula for the period of a pendulum T = 2π√(L/g), where L is the length of the pendulum and g is the acceleration due to gravity (approximately 9.8 m/s2), we can find the individual periods. Starting with the 1.44-meter pendulum (Pendulum A) and the 1-meter pendulum (Pendulum B), we calculate their periods as follows:
TA = 2π√(1.44/9.8) and TB = 2π√(1/9.8).
To find after how many vibrations they will swing together, we look for a common multiple between the number of vibrations. If Pendulum A makes n cycles and Pendulum B makes m cycles, then n x TA should equal m x TB. We look for the least common multiple (LCM) of the periods reduced to their simplest fraction form to find the point where they align again.
Since this is a simplified explanation, actual calculations would require a mathematical approach to find the exact number of vibrations for both pendulums to be in sync. This will be the LCM of the ratios of their periods.