# Scaling factor of a simple pendulum between length and time period.

## Homework Statement

The time period of a simple pendulum is doubled when the length of the pendulum is increased by 3.0m. What is the original length of the pendulum?

## Homework Equations

T= 2∏√(l/g)
also l is original length and l+3 is the new length

## The Attempt at a Solution

So
2T=2(2∏√(l/g)
2T=2∏√(4l/g)
The new length is 4 times the original length and it also has to be the original length plus 3m.
∴ 4l=l+3
so l=1m

I am happy with the mathematical steps, and it gives me the correct answer.
It is the reason why I can do this I could do with some guidance on.
Is it because when I double T the only thing that could change is the length?
Therefore I can put the multiplying factor of 2 inside the sqrt as 4×l.

## Answers and Replies

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tiny-tim
Homework Helper
hi mrcotton!
It is the reason why I can do this I could do with some guidance on.
Is it because when I double T the only thing that could change is the length?
Therefore I can put the multiplying factor of 2 inside the sqrt as 4×l.
yes, that's exactly right …

T = 2π√(l/g) means that T is proportional to √l, so twice T means 4 times l (if everything else is the same)

(and your proof is completely correct)

rude man
Homework Helper
Gold Member
What you did is fine. But you were lucky to have been able to sneak the "2" inside the square root sign so easily to make 4.

I would have said

T = 2π√(l/g)
2T = 2π√[(l+3)/g]
so 2 = √[(l+3)/l]
& solve for l.

tiny-tim
Homework Helper
isn't that the same?

Thanks ever so much for the help with getting my old grey matter working.
So in general I can solve it like this.

Did the sqrt of 4 only work for those values.
I tried doing the calculation both ways for a pendulum thats time period has tripled when the original length is extended by 6m and got the same results.

rude man
Homework Helper
Gold Member
Yeah, maybe I was a bit severe here ... did you try for new l = 1.73l? This is a very trivial point, should probably not even have raised it.

Hi Rude man, please I like the trivial points. Its the pedantic points in physics that make it interesting to me. Each physics question seems to open up more questions.

rude man
Homework Helper
Gold Member
Hi Rude man, please I like the trivial points. Its the pedantic points in physics that make it interesting to me. Each physics question seems to open up more questions.
OK I got plenty of those ...