Two simple pendulum each of length 5cm and time period of 4 second and 6 second starts oscillating one from left extreme and the other from right extreme. When will the two bodies be at the same phase?
To determine when two simple pendulums will be at the same phase, we need to find the point in their oscillations where they align.
Given that the time period of the first pendulum is 4 seconds and its length is 5 cm, we can use the formula for the time period of a simple pendulum, T = 2π√(L/g), where T is the time period, L is the length of the pendulum, and g is the acceleration due to gravity (approximately 9.8 m/s²).
Let's convert the length of the first pendulum to meters: 5 cm = 0.05 m.
Using the formula, we have:
T₁ = 2π√(0.05/9.8)
T₁ ≈ 1.414 seconds
Similarly, for the second pendulum with a length of 5 cm and a time period of 6 seconds:
T₂ = 2π√(0.05/9.8)
T₂ ≈ 1.677 seconds
Now, we want to find when the two pendulums will be at the same phase. The phase refers to the position of a pendulum in its oscillation cycle at a given time.
Since the first pendulum starts from the left extreme, it will be at the same phase as the second pendulum when it is also at the left extreme. At this point, both pendulums will have completed an integer number of oscillations.
We can determine the number of oscillations completed by each pendulum by dividing the total time elapsed by the time period of each pendulum.
For the first pendulum, the time taken to reach the left extreme is T₁/4, as its time period is 4 seconds.
Similarly, for the second pendulum, the time taken to reach the right extreme is T₂/6, as its time period is 6 seconds.
To find when both pendulums will be at the same phase, we need to determine the point where the number of oscillations for each pendulum is the same.
Since we consider the whole number of oscillations, the time when both pendulums will be at the same phase can be calculated using the least common multiple (LCM) of the two times.
LCM(T₁/4, T₂/6) = LCM(1.414/4, 1.677/6)
By simplifying and calculating the LCM, we find:
LCM(0.3535, 0.2795) = 1.5865 seconds
Therefore, the two pendulums will be at the same phase after approximately 1.5865 seconds (or 1.59 seconds, rounded to two decimal places) from the starting point.