In the arrangement shown below, an object can be hung from a string $($with linear mass density $=0\mathrm{.}00200k\frac{g}{m})$ that passes over a light pulley. The string is connected to a vibrator $($of constant frequency $f),$ and the length of the string between point $P$ and the pulley is $L=1\mathrm{.}80m.$ When the mass m of the object is either $25\mathrm{.}0$ kg or $36\mathrm{.}0$ kg $,$ standing waves are observed; no standing waves are observed with any mass between these values, however.

m

$($a$)$ What is the frequency of the vibrator $($in Hz $)?($Note: The greater the tension in the string, the smaller the number of nodes in the standing wave.$)$

Assume there are $n$ nodes when mass $36\mathrm{.}0$ kg is used and $n+1$ nodes when mass $25\mathrm{.}0$ kg is used. Hz

$($b$)$ What is the largest object mass $($in kg $)$ for which standing waves could be observed?

How is the velocity of the wave on the string related to the frequency of the wave and its wavelength? kg

$($c$)$ What If$?$ What would the linear mass density of the string have to be $($in $k\frac{g}{m})$ if $36\mathrm{.}0$ kg is the largest mass for which standing waves are observed?

$k\frac{g}{m}$

$($d$)$ For what values of $m($in kg $)$ would standing waves with the next four higher numbers of nodes be observed in this case?

$m_2=,,kg$

$m_3=,,kg$

$m_4=,,kg$

$m_5=,,kg$