Question

A wheel $4m$ in diameter rotates about a fixed frictionless horizontal axis, about which its moment of inertia is $10kg{m}^2$ A constant tension of $40N$ is maintained on a rope wrapped around the rim of the wheel. If the wheel starts from rest at $t=0s$, find the length of rope unwound till $t=3s$.

• A. $36.0m$
  
• B. $72.0m$
  
• C. $18.0m$
  
• D. $720m$
  

Answer 1

The correct option is B $72.0m$

$\begin{array}{cc}\text{Given}-& \ast \text{Diameter of wheel}=4m;\text{Radius}=2m\\ & \ast \text{Moment of Inertia}=10kg{m}^2\\ & \ast \text{Tension in Rope}=40N\end{array}$

$\begin{array}{c}\text{By torque moment equation}\\ \begin{array}{cc}\tau =I\alpha & \alpha \to \text{angular acceleration}\\ & I\to \text{Moment of Inertia.}\\ & 40N\times 2m=10\times \alpha \\ \Rightarrow & \alpha =8\text{rad}{18}^2\end{array}\end{array}$

$\begin{array}{c}\text{By equation of circular motion -}\\ \begin{array}{c}\theta ={\theta }_0+\frac{1}{2}\alpha t^2\\ \text{From}t=0\text{to}t=38\\ \theta =\frac{1}{2}\times 8\times (3{)}^2=36\text{radians.}\end{array}\\ \begin{array}{c}\text{Then arc subtended is the length of the Rope}\\ \text{Arc}=\theta \cdot R=36\end{array}=72m\text{.}\end{array}$