Question

The frequency $v$ of vibrations of uniform string of length $l$ and stretched with a force $F$ is given by
$v=\frac{p}{2 l} \sqrt{\frac{F}{m}}$
where $p$ is the number of segments of the vibrating string and $m$ is constant of the string. What is the dimensions of $m ?$

• A. $\left[ ML ^{-1} T ^{-1}\right]$
• B. $\left[ ML ^{-3} T ^{0}\right]$
• C. $\left[ ML ^{-2} T ^{0}\right]$
• D. $\left[ ML ^{-1} T ^{0}\right]$

Answer 1

Answer: (D)

Squaring both sides of the given relation, we get
$v^{2}=\frac{p^{2}}{4 l^{2}} \frac{F}{m}$
or $m=\frac{p^{2} F}{4 l^{2} v^{2}}$
$\therefore $ dimensions of $m$
$=\frac{\text { dimensions of } F}{\text { dimensions of } l^{2} \times \text { dimensions of } v^{2}}$
($\because p$ is a dimensionless quantity)
$=\frac{\left[ MLT ^{-2}\right]}{\left[ L ^{2}\right]\left[ T ^{-1}\right]^{2}}$
$=\left[ ML ^{-1} T ^{0}\right]$