Question

Verify if this equation is true based on physics dimensions :

T = 2 pi sqrt(l/g)

As T = time, l = length , g = gravity

Answer 1

hint:
we have:
\[\sqrt {\frac{l}{g}} = \sqrt {\frac{{{\text{meters}}}}{{{\text{meters/second}}{{\text{s}}^{\text{2}}}}}} \]

Answer 2

I've solved it and got T^-1 as the dimension.
The problem is , I can't find which answer is correct
a) correct b) correct according to dimensions c) wrong d) wrong according to dimensions

d or c ? and why ?

Answer 3

it is correct! namely option a)

Answer 4

why ?

Answer 5

Ops

Answer 6

a good method is to start from the differential equation of the motion of the pendulum

Answer 7

m/m/s^2 = sqrt(s^2)

Answer 8

But why b is wrong ?

Answer 9

yes! it is correct, nevertheless your reasoning is not able to justify the presence of the coefficient \(2 \pi\)
option b) is wrong since we have the correct formula, and the dimensional analysis can not say anything on possible numerical coefficients which may accompany the various formulas

Answer 10

we cannot check if 2 pi is correct therefore b is correct

Answer 11

in other words, using the dimensional analysis I am able to write this:
\[T \propto \sqrt {\frac{l}{g}} \]

Answer 12

Yeah, so correct according to dimension ?

Answer 13

no, since you have this formula:
\[T = 2\pi \sqrt {\frac{l}{g}} \]

Answer 14

a or b ?

Answer 15

I think it is option a)

Answer 16

Doesn't a mean "We are fully sure that the equation dimensions are correct, constants are correct" ?

Answer 17

this formula:
\[T \propto \sqrt {\frac{l}{g}} \]
is correct according to dimensions

Answer 18

so if the constant is not written it's correct according to dimensions
if it's written , just correct ?

Answer 19

yes!

Answer 20

THANKS.