Name: Shear Modulus
Uploaded: 2019-07-19
Duration: 7 min 26 s
Description: In this video, we will discuss the most important and interesting property of a material that tells us how resistance a material which undergone shear deform...

Question

A steel wire of cross-section

A

is stretched horizontally between two clamps located at a distance

l

from each other. A load

W

is suspended from the mid-point of the wire. What is the vertical distance through which the mid-point of the wire moves down ? Young's modulus of steel is

Y

.

Accepted Answer

Here change in length is $△ l = (AC + BC) - 2 l$

$∆ l = 2 {(l^{2} + x^{2})}^{\frac{1}{2}} - 2 l$

$∆ l = 2 l {(1 + \frac{x^{2}}{l^{2}})}^{\frac{1}{2}}$

$∆ l = \frac{x^{2}}{l^{2}}$

$∴ stress = \frac{∆ l}{2 l} = \frac{x^{2}}{2 l^{2}}$

Now, If T is the tension in the string, then

$2 Tcosθ = W$

$T = \frac{W}{2 cosθ}$

$cosθ = \frac{x}{OB} = \frac{x}{\sqrt{l^{2} + x^{2}}}$

$cosθ = \frac{x}{l {(1 + \frac{x^{2}}{l^{2}})}^{\frac{1}{2}}} = \frac{x}{l [1 + \frac{x^{2}}{2 l^{2}}]}$

As, $\frac{x^{2}}{2 l^{2}} < < l$

So, $l + \frac{x^{2}}{2 l^{2}} = 1$

$∴ cosθ = \frac{x}{l}$ And $T = \frac{W}{2 (\frac{x}{l})} = \frac{Wl}{2 x}$

$∴ stress = \frac{T}{A} = \frac{Wl}{A 2 x}$

Now, Young's modulus of steel is $Y = \frac{Stress}{strain} = \frac{Wl}{2 Ax} \times \frac{2 l^{2}}{x^{2}} = \frac{{Wl}^{3}}{{Ax}^{3}}$

$∴ x = l {[\frac{W}{YA}]}^{\frac{1}{3}}$

OR $x = \frac{l}{2} {(\frac{W}{AY})}^{\frac{1}{3}}$

A steel wire of cross-section $A$ is stretched horizontally between two clamps located at a distance $l$ from each other. A load $W$ is suspended from the mid-point of the wire. What is the vertical distance through which the mid-point of the wire moves down ? Young's modulus of steel is $Y$ .

Important Questions on Elasticity

Learn and Prepare for any exam you want !

A steel wire of cross-section A is stretched horizontally between two clamps located at a distance l from each other. A load W is suspended from the mid-point of the wire. What is the vertical distance through which the mid-point of the wire moves down ? Young's modulus of steel is Y.

Important Questions on Elasticity

Learn and Prepare for any exam you want !

A steel wire of cross-section $A$ is stretched horizontally between two clamps located at a distance $l$ from each other. A load $W$ is suspended from the mid-point of the wire. What is the vertical distance through which the mid-point of the wire moves down ? Young's modulus of steel is $Y$ .