Rotation Ques 1
1 A uniform rod of length $l$ is being rotated in a horizontal plane with a constant angular speed about an axis passing through one of its ends. If the tension generated in the rod due to rotation is $T(x)$ at a distance $x$ from the axis, then which of the following graphs depicts it most closely?
(2019 Main, 12 April I)
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Answer:
Correct Answer: 1.( b )
Solution:
To find tension at $x$ distance from fixed end, let us assume an element of $d x$ length and $d m$ mass. Tension on this part due to rotation is
As,
$ \begin{aligned} d T & =K x \\ K & =m \omega^2 \\ K & =(d m) \omega^2 \\ d T & =(d m) \omega^2 x \end{aligned} $
To find complete tension in the rod, we need to integrate Eq. (iii),
$ \int_0^T d T=\int_0^m(d m) \omega^2 x $
Using linear mass density,
$ \begin{aligned} \lambda & =\frac{m}{l}=\frac{d m}{d x} \\ \Rightarrow \quad d m & =\frac{m}{l} \cdot d x \end{aligned} $
Putting the value of Eq. (v) in Eq. (iv), we get
$ \begin{aligned} T & =\int_x^l \frac{m}{l} \cdot \omega^2 x \cdot d x=\frac{m}{l} \cdot \omega^2\left[\frac{x^2}{2}\right]_x^l \\ \Rightarrow \quad T & =\frac{m \omega^2}{2 l}\left[l^2-x^2\right] \text { or } T \propto-x^2 \end{aligned} $
BETA
