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

ndnordic | Certified Educator

calendarEducator since 2009

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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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enotechris | Certified Educator

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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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plantcell | Student

I am not an expert at physics but to put it in simple terms:

By increasing the length of pendulum, it increases the length of the arc movement by the pendulum. (arc length is given by angle/360 multiplied by 2(pi) and r; in this case r is your length of pendulum) Since g is constant, velocity is constant. Time duration for 1 oscillation hence increases.

@user922554, to reply your comment (with reference to the 1st answer), there is no difference in potential or kinetic energy, just the difference in the distance the pendulum bob travelled. Perhaps I can only guess that you are talking about how energy is transferred as time passes, so, as the bob is swinging, a bob with a longer pendulum length will have a lower rate of energy transfer, but there should not be a difference in maximum kinetic energy reached. (This is all very confusing without drawing actual setups)

Consider 2 setups, where the only difference in the setups is the length of the pendulum, the height above ground when the bob is released is kept constant, starting swing from right to left. Assuming no loss in heat or sound.

To make it simple,

Results 1st setup: when t=0, original height(right side), t=1, lowest height, t=2, same height as t=0, t=3 lowest height, t=4 back to original height.

Results 2nd setup: t=0, original, t=2 lowest, t=4 same as t=0, t=6 lowest, t=8 back to original.

(Lets ignore the fact that I am using hypothetical experiment to explain this), at t=1, for the 1st setup, the bob has maximum velocity as it reaches the lowest height from original height. For the 2nd setup, the bob has converted half of the maximum energy possible, ke = 1/2 mv^2, so v will vary accordingly. (However do note that maximum velocity--> t=1 for 1st setup, and t=2 for 2nd setup reached should be the same [unless I visualised wrongly])

So, increasing the length of the pendulum, increases the arc length, of which energy transfer from gpe to ke is slower. On average, the speed of different setups should be the same, just given the different arc lengths, the arc movement varies, hence if arc length increases, time taken for 1 swing increases.

Rate of energy transfer do not directly affect time period of a pendulum, energy transfer do affect the velocity of the pendulum, but *average* speed, is the same or around the same, hence a longer travelling distance will have a longer period.

shrinath | Student

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)