Do you understand dimensional analysis for physics?

Question

T = 2 pi sqrt(l / g)

How do you show that this equation is dimensionally consistent.

Answer 1

It's very easy. Your equation contains the following quantities (forget about 2pi, which have no dimension):

T [s]
L [m]
g [m/s^2]

In dimensional analysis, you just have to replace each quantity with its corresponding unit and simplify the equation:

[s] = sqrt ([m]/([m]/([s]*[s])))
[s] = sqrt (1/(1/([s]*[s])))
[s] = sqrt([s]*[s])
[s] = [s]

Q.E.D.

Answer 2

T= 2п√ (l/g)


The dimensional formula for L>H>S is
M^0* L^0*T^1

2п in the right hand side is a dimensionless constant.

The dimensional formula for √ (l ) is M^0* L^(1/2)*T^0
= L^(1/2)

The dimensional formula for √ (1/g ) is
M^0* L^( - 1/2)*T^1
=T^(1/2) /L^(1/2)

The dimensional formula for √ (l/g) is T^1

Dimensionally both sides are the same.

The simplest way is to check the units of quantities involved.

The unit of a quantity is the unit derived from the base units.

Hence the units represent the dimensions.

But while checking by units all derived units must be converted into its base units.

Example the unit of force is newton.

Convert newton to kgms^(-2)

In the equation T= 2п√ (l/g)

The L.H.S unit is second.

R.H.S unit is √ (meter / {meter/ (second)^2} = second.

Answer 3

If you divide the square root of the units of "L" by the square root of the units of "g" you should get the same units as T has.

T is period I think so seconds.

sec= (m^.5/(m^.5/sec))
sec=sec