In nature, the failure of structural members usually result from large torque because of twisting or bending rather than due to tensile or compressive strains. This process of structural breakdown is called buckling and in cases of tall cylindrical structures like trees, the torque is caused by its own weight bending the structure. Thus the vertical through the centre of gravity does not fall within the base. The elastic torque caused because of this bending about the central axis of the tree is given by $\frac{\mathit{Y}\mathit{\pi }{\mathit{r}}^4}{4\mathit{R}}$ . Y is the Young’s modulus, r is the radius of the trunk and R is the radius of curvature of the bent surface along the height of the tree containing the centre of gravity (the neutral surface). Estimate the critical height of a tree for a given radius of the trunk.

This is a long answer type question as classified in NCERT Exemplar

Consider the diagram according, the bending torque on the trunk of radius r of the tree = $\frac{Y\pi r^4}{4R}$

When the tree is about to buckle Wd= $\frac{Y\pi r^4}{4R}$

If R>>h, then the centre of gravity is at a height l $\approx \frac{h}{2}\mathrm{f}\mathrm{r}\mathrm{o}\mathrm{m}\mathrm{}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{}\mathrm{g}\mathrm{r}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}$

From $?ABC{R}^2\approx (R-d)$ ²+ (h/2)²

If d <2+

So d = h²/8R

If w_o is the weight /volume

$\frac{Y\pi r^4}{4R}=w_o\left(\pi r^{2}h\right)\frac{h^2}{8R}$

h= ( $\frac{2Y}{w_o}$ )^1/3r^2/3

critical height = h= ( $\frac{2Y}{w_o}$ )^1/3r^2/3

<p><span data-teams="true">This is a long answer type question as classified in NCERT Exemplar</span></p><div><div><p>Consider the diagram according, the bending torque on the trunk of radius r of the tree =<span contenteditable="false">&nbsp;<math><mfrac><mrow><mrow><mi>Y</mi><mi>π</mi><msup><mrow><mrow><mi>r</mi></mrow></mrow><mrow><mrow><mn>4</mn></mrow></mrow></msup></mrow></mrow><mrow><mrow><mn>4</mn><mi>R</mi></mrow></mrow></mfrac></math></span></p><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1745489974phpOqDoEP_480x360.jpeg" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1745489974phpOqDoEP.jpeg" alt="" width="215" height="135"></picture></div><div><p>When the tree is about to buckle Wd=<span contenteditable="false">&nbsp;<math><mfrac><mrow><mrow><mi>Y</mi><mi>π</mi><msup><mrow><mrow><mi>r</mi></mrow></mrow><mrow><mrow><mn>4</mn></mrow></mrow></msup></mrow></mrow><mrow><mrow><mn>4</mn><mi>R</mi></mrow></mrow></mfrac></math></span></p><p>If R&gt;&gt;h, then the centre of gravity is at a height l<span contenteditable="false">&nbsp;<math><mo>≈</mo><mfrac><mrow><mrow><mi>h</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></mfrac><mi mathvariant="normal">f</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">m</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">t</mi><mi mathvariant="normal">h</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal"></mi><mi mathvariant="normal">g</mi><mi mathvariant="normal">r</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">u</mi><mi mathvariant="normal">n</mi><mi mathvariant="normal">d</mi></math></span></p><p>From<span contenteditable="false">&nbsp;<math><mo>? </mo><mi>A</mi><mi>B</mi><mi>C</mi><msup><mrow><mrow><mi>R</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup><mo>≈</mo><mo> (</mo><mi>R</mi><mo>-</mo><mi>d</mi><mo>)</mo></math>&nbsp;</span>²+ (h/2)²</p><p>If d &lt;2+</p><p>So d = h²/8R</p><p>If w_o&nbsp;is the weight /volume</p><p><span contenteditable="false"><math><mfrac><mrow><mrow><mi>Y</mi><mi>π</mi><msup><mrow><mrow><mi>r</mi></mrow></mrow><mrow><mrow><mn>4</mn></mrow></mrow></msup></mrow></mrow><mrow><mrow><mn>4</mn><mi>R</mi></mrow></mrow></mfrac><mo>=</mo><msub><mrow><mrow><mi>w</mi></mrow></mrow><mrow><mrow><mi>o</mi></mrow></mrow></msub><mo> (</mo><mi>π</mi><msup><mrow><mrow><mi>r</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup><mi>h</mi><mo>)</mo><mfrac><mrow><mrow><msup><mrow><mrow><mi>h</mi></mrow></mrow><mrow><mrow><mn>2</mn></mrow></mrow></msup></mrow></mrow><mrow><mrow><mn>8</mn><mi>R</mi></mrow></mrow></mfrac></math></span></p><p>h= (<span contenteditable="false">&nbsp;<math><mfrac><mrow><mrow><mn>2</mn><mi>Y</mi></mrow></mrow><mrow><mrow><msub><mrow><mrow><mi>w</mi></mrow></mrow><mrow><mrow><mi>o</mi></mrow></mrow></msub></mrow></mrow></mfrac></math>&nbsp;</span>)^1/3r^2/3</p><p>critical height = h= (<span contenteditable="false">&nbsp;<math><mfrac><mrow><mrow><mn>2</mn><mi>Y</mi></mrow></mrow><mrow><mrow><msub><mrow><mrow><mi>w</mi></mrow></mrow><mrow><mrow><mi>o</mi></mrow></mrow></msub></mrow></mrow></mfrac></math>&nbsp;</span>)^1/3r^2/3</p></div></div></div></div>

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