Answer to Question #135304 in Mechanics | Relativity for Sofia rose

Question

A student taking a quiz finds on a reference sheet the two equations f = 1 / T and =square root of (1/T).
She has forgotten what T represents in each question.
Use dimensional analysis to determine the units required for T in each question.
Explain how you can identify the physical quantity each T represents from the units.

Answer 1

The reciprocal of period of oscillation is the frequency

$f =\dfrac{1}{T}$

f is frequency and T is period

Rearranging the equation for T

$T =\dfrac{1}{f}$

using the dimensional analysis to write units for period

$[T] =\dfrac{1}{[f]}$

substituting dimension formula T^-1 for f

$[T] =\dfrac{1}{[\left (T^{\smash{-1}}\right)]}$

=T

where T is time

$\therefore$ units for T are seconds

The expression for velocity of wave is,

$v=\sqrt{\dfrac{T}{\mu}}$

where T is tension in the string , and $\mu$ is mass per unit length.

Rearranging the equation for T

$T=\mu*v^2$

using dimensional analysis to write units for the period

$[T]=[\mu]*[v^2]$

substituting dimensional formula $\frac{M}{L}$ for $\mu$ and $\frac{L^2}{T^2}$ for v.

$[T]=\frac{M}{L} \frac{L^2}{T^2}$

$=\frac{ML}{T^2}$

Here M represents mass, L represents length, and T is time

Therefore, the units for tension is kg.m/s² or newtons

b). In the first part, T represents period of time . in the second part of the solution . T represents the force of tension.