Torsional Rigidity of a Cylinder and Bending Moment of a Beam

\begin{align*}\rightarrow d\tau =\eta \frac{2\pi \theta }{l}x^3dx \rightarrow 2 \end{align*} \(\rightarrow \)A uniform rod having length much greater than its thickness is called a beam. Bending moment, \(B=\sum \frac{Y}{R}ax^2\)\begin{align*}=\frac{Y}{R}\sum ax^2 \end{align*}

Summary

Things to Remember

\(\rightarrow \)For solid cylinder, \(c=\frac{\tau }{\theta }=\frac{\eta \pi R^4}{2l}\)
For hollow cylinder, \(c=\frac{\tau }{\theta }=\eta \frac{\pi (R^4-r^4)}{2l} \)

\(\rightarrow \) For a beam of rectangular cross-section having breadth 'b' and thickness 't',\(I=\frac{bt^3}{12} \)

For the beam having circular cross-section \(I=\frac{\pi R^4}{4} \)

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Q1:

Write short notes on mycology.

Type: Short Difficulty: Easy

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Answer: <h4>Introduction</h4>
Mycology is the branch of biology concerned with the study of fungi, including their genetic and biochemical properties, their taxonomy and their use to humans as a source for tinder, medicine (e.g. penicillin), foods (beer, wine, cheese, mushrooms) and entheogens, as well as their dangers, such as poisoning or infection.
<ul>
<li>The fungus is a Latin word that means “mushroom”.</li>
<li>Branch of biology that deals with a study of fungi is termed as Mycology.</li>
<li>Medical Mycology is a study of fungal epidemiology, ecology, pathogenesis, diagnosis and treatment in human beings.</li>
<li>Greek “myces” means mushroom.</li>
</ul>
Fungi are “Eukaryotic, a spore-bearing achlorophyllous organism that may reproduce sexually and asexually and whose filamentous, branched and somatic structure are typically surrounded by cell wall containing chitin, cellulose or both of this substance with many other complex carbohydrates.”
<ul>
<li>They are unicellular or multicellular</li>
<li>The main component of cell wall is chitin</li>
<li>Fungi are not affected by antibiotics due to a presence of chitin instead of peptidoglycan.</li>
<li>Instead of cholesterol, the cell membrane is composed of ergosterol.</li>
<li>Saprophytic and parasitic in nature.</li>
<li>Lack of chlorophyll and chemolithotrophic machinery.</li>
<li>Heterotrophic in nature, require external carbon source.</li>
<li>Produce enzyme that digests & then absorb nutrient.</li>
<li>Circulation of nutrient takes place with a flow of protoplasm.</li>
<li>Reproduce sexually or asexually with a production of spores.</li>
<li>Broadly divided into two main groups.</li>
</ul>
-Molds (filamentous form)
-Yeasts (nonfilamentous form)
Filamentous fungi may form a cross wall between each cell called septa. They may be septate or aseptate or coenocytic.
 
 
 
 
<h4>Fungal structure and mycology</h4>
<ol>
<li>Capsule</li>
<li>Cell wall</li>
<li>Plasmalemma</li>
<li>Cytoplasmic content\structure</li>
</ol>
 
<img src="http://images.slideplayer.com/13/4015100/slides/slide_39.jpg" alt="Image result for Structure of yeast" width="365" height="274" />
Figure: Structure of yeast
 
<img src="http://www.brc.hu/pictures/bk_nagy_fig1.png" alt="Image result for Structure of Filamentous Fungi" width="381" height="148" />
Figure: Structure of Filamentous Fungi
<ol>
<li>Capsule</li>
</ol>
<ul>
<li>Some fungi are Cryptococcus neoformans produce external slime or more compact capsule.</li>
<li>Composed predominantly of amorphous polysaccharide that may be mucilaginous and cause cell adhere and clump together.</li>
<li>Do not affect the permeability of cell wall and membrane.</li>
<li>Determines virulence.</li>
<li>Plays an important role in eliciting host immune response.</li>
<li>Demonstration of the capsule by(Distinct halo around the yeast cells is twice the size of a round yeast cell).</li>
</ul>
-India ink preparation.
-India ink with 2 % chromium mercury(both internal and external structure).
-Nigrosin with formalin.
 
2. Cell wall
<ul>
<li>Composed of 15 -18% of dry weight of fungi.</li>
<li>Provide rigidity and strength.</li>
</ul>
Fig: Structure of cell wall
<ul>
<li>Protect from osmotic shock.</li>
<li>Thicker in yeast(200-300 nm) than mold(200nm).</li>
<li>Lie exterior to plasmalemma.</li>
</ul>
Composition
<ul>
<li>80% carbohydrate.</li>
<li>10% protein and glycoprotein.</li>
<li>Wall protein concentration high near the membrane.</li>
<li>Contain a large amount of sulphur containing amino acid and disulphide bond àHigher in hyphae than yeast.</li>
<li>Reduction in disulphide bond is associated with mycelia to yeast morphogenesis.</li>
</ul>
 
Yeast cell wall
 
<img src="http://www.allaboutfeed.net/PageFiles/21092/fg1.jpg" alt="Image result for Structure of yeast cell wall" />
Figure: Structure of yeast cell wall
 
Importance of cell wall in pathogenic fungi
<ul>
<li>Protect fungal cell from external injuries.</li>
<li>Mediates interaction with host cell; adhesion, colonization, signaling and immune recognition.</li>
<li>It acts as a protective barrier as is an obstacle that must be considered for choosing antifungal for potential effectiveness.</li>
<li>Cell wall being unique in chitin and β-glucan being absent in host can act as a site for antifungal.</li>
<li>Inhibition of cell wall causes death or lysis.</li>
</ul>
 
3. Plasmalemma
<ul>
<li>Phospholipid layer.</li>
<li>Membrane inner to cell wall that encloses the complex cytoplasmic content.</li>
<li>Regulates the intake and secretion of solute.</li>
<li>Composed of several phospholipids, most prevalent phosphatidyl glycerol.</li>
<li>Principal fungal sterol à ergosterol and zymosterol.</li>
<li>Lipid consists 30% dry weight.</li>
<li>Facilitate the synthesis of a cell wall and capsular component.</li>
<li>Also the site for β-glucan synthesis. Ergosterol instead of cholesterol helps in anti-fungal strategy.</li>
</ul>
 
4. Cytoplasmic structures
Consists of a nucleus, nuclear membrane, mitochondria, micro vesicles, microfilament, ribosome (80s),Golgi apparatus, double membrane endoplasmic reticulum, vacuoles, and other cytoplasmic structure.
 
Nucleus
<ul>
<li>Nuclei enclosed of nuclear membrane with a unique property that persists throughout the metaphase of mitotic cycle.</li>
<li>Consist of linear circular chromosomal DNA.</li>
<li>True nucleoli with RNA.</li>
</ul>
 
Mitochondria
<ul>
<li>Consist of flat cristae.</li>
<li>The number varies per cell that correlates with respiratory activity.</li>
<li>Sporulation à decrease.</li>
<li>Mold to yeast morphogenesis à increase.</li>
</ul>
 
Microfilaments
<ul>
<li>Function as supporting framework.</li>
<li>Cytoplasmic skeleton composed of these microfilaments os actin and tubulin- containing microtubules.</li>
</ul>
Vacuoles
<ul>
<li>Contain various hydrolytic enzymes.</li>
<li>Storage of ion, metabolites as amino acids, polyphosphate, and another compound.</li>
</ul>
Storage granules
<ul>
<li>Storage of energy in the form of lipids and glycogen.</li>
</ul>
Secretory vesicles
<ul>
<li>Vesicles present in a hyphal tip contains enzyme like chitin synthase and other synthetic enzymes, lytic enzyme, or co-factor as well as cell wall precursor.</li>
</ul>
Secondary metabolites
<ul>
<li>Carcinogen (aflatoxin), toxin (amanitin), antibiotics, anti-cancer substances, pharmacologically active compound (ergotamine).</li>
<li>Production of killer toxins by Saccharomycetes cerevisal, Candida & Cryptococcus.</li>
</ul>

Videos

Introduction to Mycology

Torsional Rigidity of a Cylinder and Bending Moment of a Beam

Torsional Rigidity of a Cylinder

Consider a cylinder of length'l' and radius R such that its upper end is fixed to a rigid support and at the lower end a twisting couple is applied in plane perpendicular to its length.
Consider the cylinder to be composed of large number of co-axial elementary cylinders. Consider one of such cylinders having radius x and thickness dx. The cross section PQSR now becomes parallelogram PQ'S'R as shown in figure 3, where shearing angle <QPQ'=\(\phi \)
from figure 2, \(\theta =\frac{QQ'}{x} \implies QQ'=\theta x\)
Again, from figure 3, \(\phi =\frac{QQ'}{PQ}=\frac{\theta x}{l} \)
Now, the modulus of rigidity of the material of the cylinder is \begin{align*}\eta =\frac{F/A}{\phi }\implies F=\eta \phi A \rightarrow 1 \end{align*}where, \(A=2\pi xdx\) is the cross-sectional area of the elementary cylinder.\begin{align*}\implies F=\eta \phi 2\pi xdx \end{align*}Now, moment of this force on the elementary cylinder about OO' is \begin{align*}d\tau =Fx=\eta \phi 2\pi x^{2}dx \end{align*}\begin{align*}=\eta (\frac{\theta x}{l} )2\pi x^{2}dx \end{align*}\begin{align*}d\tau =\eta \frac{2\pi \theta }{l}x^3dx \rightarrow 2 \end{align*}

Case I: For a solid cylinder

In case of solid cylinder, the elementary co-axial cylinder vary from x=0 to x=R. Thus from 2,\begin{align*}\tau =\int_{o}^{R}d\tau \end{align*}\begin{align*}\int_{o}^{R}\eta \frac{2\pi \theta }{l}x^3dx \end{align*}\begin{align*}\implies Twisting\space couple,\space \tau=\eta \frac{\pi \theta R^4}{2l} \end{align*}
Twisting couple per unit twisting angle is called torsional rigidity (c)\begin{align*}i.e.\space c=\frac{\tau }{\theta }=\frac{\eta \pi R^4}{2l} \rightarrow 3 \end{align*}

Case II: For a hollow cylinder

In case of hollow cylinder, the elementary co-axial cylinders vary from x=r to x=R. Thus from 2\begin{align*}\tau =\int_r^R \eta \frac{2\pi \theta }{l}x^3dx \end{align*}
Twisting couple, \(\tau =\eta \frac{\pi \theta (R^4-r^4)}{2l}\)
Twisting couple per unit twisting angle is called torsional rigidity(c) i.e.
\begin{align*}torsional\space rigidity,\space c=\frac{\tau }{\theta }=\eta \frac{\pi (R^4-r^4)}{2l}\rightarrow 4 \end{align*}

Bending Moment of a Beam

A uniform rod having length much greater than its thickness is called a beam. When a beam is subjected to two equal; but opposite torques at the two ends, a part of the beam undergoes elongation and becomes convex and other part undergoes compression and becomes concave.

In between these two surfaces there lies a surface which is neither elongated nor compressed called neutral surface.Consider a small piece of a beam ABCD of length L before bending. Also, let after bending, the vertices become A'B'C'D' with the neutral surface having length L. The two cross-sectional sides A'D' and B'C' are produce to meet at O with an angle \(\theta \) between A'O and B'O.
Let PO=QO=R be the radius of curvature of normal surface. Let us take any one of the several possible elementary strips 'MN' at a distance x from the neutral surface, having cross-section 'a'. Then, from figure,

MN=L+l where, l=extension produced

MO=NO=R+x

From figure, considering the arc MN,\begin{align*}\theta =\frac{MN}{MO}=\frac{L+l}{R+x} \implies L+l=R\theta +r\theta \rightarrow 2 \end{align*}Substituting L from 1 to 2,\begin{align*}l=x\theta \rightarrow 3 \end{align*}Dividing 3 by 1\begin{align*}\frac{l}{L}=\frac{x}{R} \end{align*}i.e. longitudinal strain on the surface MN\(=\frac{l}{L}=\frac{x}{R}\)

Now, the Young's Modulus of elasticity of the material of the beam is\begin{align*}Y=\frac{\frac{F}{A}}{\frac{l}{L}} \end{align*}\begin{align*}\implies F=Y.\frac{x}{R}.a \end{align*}The moment of this force on the elementary strip 'MN' about the neutral surface is given by

Moment of force MN\begin{align*}=Fx \end{align*}\begin{align*}=Y.\frac{x^2}{R}.a \end{align*}

Now the bending moment of the beam is given by the algebric sum of moment on the elementary surfaces i.e.

Bending moment, \(B=\sum \frac{Y}{R}ax^2\)\begin{align*}=\frac{Y}{R}\sum ax^2 \end{align*}\begin{align*}\implies B=\frac{Y}{R}I_g \end{align*}where \(I_g=\sum ax^2\) is the geometrical moment of inertia of the cross-section of beam also called second moment.

Case I: For a beam of rectangular cross-section having breadth 'b' and thickness 't'.

\begin{align*}I=\frac{bt^3}{12} \end{align*}

Case II: For the beam having circular cross-section

\begin{align*}I=\frac{\pi R^4}{4} \end{align*}

References

Adhikari, Pitri Bhakta. A Textbook of Physics Volume-I. Kathmandu: Sukunda Pustak Bhawan, 2015.

Feynman, Richard P. The Feynman Lectures on Physics Volume 1. Noida: Dorling Kindersley (India) Pvt. Ltd., 2014.

Mathur, D S. Mechanics. New Delhi: S. Chand & Company Pvt. Ltd., 2015.

Young, Hugh D, Roger A Freedman and A Lewis Ford. University Physics. Noida: Dorling Kindersley (India) Pvt. Ltd., 2014.

Lesson

Elasticity

Subject

Physics

Grade

Bachelor of Science

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