91Ó°ÊÓ

i) Total cars are 36.

ii) Each car carries 60 passengers.

iii) Diameter of circle is$\text{d}=76\text{″¾}$.

iv) Time is $\text{T}=2\text{″¾in}$.

Use the concept of work done related to change in angular kinetic energy.First, find the angular velocity and moment of inertia.Thenplugging it in the equation, find the work done.Also, find the range of work done by plugging the average value of mass of the person.

Formulae:

$\text{W}=\frac{1}{2}\text{I}{\mathrm{Ó\neg }}_f^2−\frac{1}{2}\text{I}{\mathrm{Ó\neg }}_i^2$

$\text{I}={\text{MR}}^2$

$\mathrm{Ó\neg }=\frac{2\mathrm{Ï€}}{\text{T}}$

Amount of work required of the machinery to rotate the passengers alone:

First, find the total number of people in 36 cars.

$36×60=2160$

Angular velocity can be found from time

$\begin{array}{c}\mathrm{Ó\neg }=\frac{2\mathrm{Ï€}}{\text{T}}\\ =\frac{2×3.14}{120}\\ =0.05\text{ r²¹»å}/\text{s}\end{array}$

Initiallytheangular velocity is zero, so we can write,

role="math" localid="1661513224896" $\begin{array}{c}\text{W}=\frac{1}{2}\text{I}{\mathrm{Ó\neg }}_f^2−\frac{1}{2}\text{I}{\left(0\right)}^2\\ =\frac{1}{2}\text{I}{\mathrm{Ó\neg }}_f^2\end{array}$

Radius of circle is,

$\begin{array}{c}\text{R}=\frac{d}{2}\\ =\frac{76}{2}\\ =38\text{″¾}\end{array}$

Moment of inertia of the wheel is$\text{I}={\text{MR}}^2$

Plugging these values in equation of W, we get

$\begin{array}{c}\text{W}=\frac{1}{2}\text{M}{\left(38\right)}^2{\left(0.05\right)}^2\\ =1.805\text{ M}\end{array}$

This is work done for one person.Forall people, we can write,

$\text{W}=2.0\text{ M}×2160$

Average mass of a person varies from 50kg to 100kg. We get the range of work done:

$2.0×50×2160≤\text{W}≤2.0×100×2160$

$2.0×{10}^5\text{J}≤\text{W}≤4.0×{10}^5\text{J}$