Physics

A uniform rod of mass m and length l is rotating with constant angular velocity w about its axis which passes through its one end and perpendicular to the length of rod. the area of cross section of rod is A and its Youngs Modulus is Y. The strain at the midpoint of the rod (neglect gravity) is

Easy Math Editor

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Comments

Hi , let us say the rod is $AB$ being rotated about $A$ , and the center of rod being $C$ .

Consider force diagram on $CB$ , having mass $\frac{m}{2}$ whose center of mass is rotating about $A$ in a circle of radius $\frac{l}{2} + \frac{l}{4} = \frac{3l}{4}$ . If the tension (providing centripetal force) is $T$ ,

$T = \frac{m}{2} \omega^2 \frac{3l}{4}$

Hence, strain = $\frac{T}{AY} = \frac{3m \omega^2 l }{8AY}$

jatin yadav - 7 years, 6 months ago

Consider an infinitesimally small element of $dr$ thickness at a distance $r$ from the axis.

Let $T(r)$ be the tension at $r$ .

Switching to a rotating reference frame and balancing the forces on this small element.

$T(r)=T(r+dr)+(dm)\omega^2r$

where $dm=(m/l)dr$ .

$\Rightarrow -T'(r)dr=(m/l)\omega^2r\,dr \Rightarrow -dT=(m/l)\omega^2r\,dr$

$\displaystyle \Rightarrow \int_{0}^r \frac{m}{l}\omega^2r\,dr=\int_{T_0}^{T} -dT$

Solving for T, we get:

$\displaystyle T=T_0-\frac{m\omega^2r^2}{2l}$

At $r=l$ , $T=0 \Rightarrow T_0=\frac{m\omega^2l^2}{2l}$ . Hence,

$\displaystyle T=\frac{m\omega^2}{2l}(l^2-r^2)$

Since we need tension at l/2,

$\displaystyle T=\frac{m\omega^2}{2l}\left(l^2-\frac{l^2}{4}\right) =\frac{3m\omega^2l}{8}$

Hence, strain= $\displaystyle \frac{T}{AY}=\frac{3m\omega^2l}{8AY}$

Pranav Arora - 7 years, 6 months ago

A particle moves in a circular path of radius R emitting a sound of frequency 'f'.what will be the maximum and minimum frequencies of sound heard by a observer standing at distance (R/2) from the centre .

Utkarsh Gupta - 2 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$