# Why does the time period of the pendulum increase when its length is increased?

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Perhaps the easiest way to understand this is to look at the arm of the pendulum as the radius of a circle. As the pendulum swings back and forth in its periodic motion it is moving through a fixed portion of the circumference of the circle. If you keep the angle of the pendulum's motion the same but increase the length of the pendulum you will also increase the length of the arc through which the pendulum is moving. Since the force acting on the pendulum is constant (gravity acting on the mass), the velocity of the pendulum is constant. Because it has to travel a longer distance, the time increases.

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This does not answer the question. A pendulum transfers energy between kinetic and potential. A true explanation must consider how the rate of transfer changes depending on the length of the pendulum.

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For a simple pendulum, where the pendulum is itself considered of negligible mass which moves with a minimal amplitude, the time period is given by the formula

T = 2(pi)*(sqrt(L/g))

where T is the time period, L is the length of the pendulum, and g is the gravitational constant.  By lengthening the pendulum, the weight at the end must travel a longer distance as it swings, which means more time is required for it to complete a given period.  See the link below for a simple pendulum calculator, enter values for different lengths and verify this for yourself:

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Lengthening a pendulum does not increase the distance that the end must travel, you completely miss the point. A 4 meter long pendulum swinging only 5cm back and forth still takes twice as long to swing 1 period than a 1 meter long pendulum swinging 10cm or 15cm back and forth. The distance of travel is irrelevant and more should be expected of a college teacher.

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The period of a simple pendulum is:2(pi)*sqrt(L/g).

So T is proportional to length.

So pendulums of same period make eaqual angle.

shorter pendulums must swing faster to travel the arc of the angle in the given period,While Big L pendulums have to travel long length(period) to mark the same arc of the angle.(logical)

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T is not proportional to Length, it is proportional to the square-root of the length (as your own equation here states). All you've done is recite an equation. This does not answer the question of "why".