# A Mild Steel Wire of Length 1.0 M and Cross-sectional Area 0.50 × 10–2 Cm2 Is Stretched, Well Within Its Elastic Limit, Horizontally Between Two Pillars. a Mass of 100 G is Suspended from the Mid-point of the Wire. Calculate the Depression at the Midpoint - Physics

A mild steel wire of length 1.0 m and cross-sectional area 0.50 × 10–2 cmis stretched, well within its elastic limit, horizontally between two pillars. A mass of 100 g is suspended from the mid-point of the wire. Calculate the depression at the midpoint.

#### Solution 1

Let AB be a mild steel wire of length 2L = lm and its cross-section area A = 0.50 x 10-2 cm2. A mass m = 100 g = 0.1 kg is suspended at mid-point C of wire as shown in figure. Let x be the depression at mid-point i.e., CD = x

:. AD = DB = sqrt((AC^2 + CD^2)) = sqrt(L^2+x^2)

:. Increase in length triangleL = (AD+DB) - AB = 2sqrt(L^2+x^2) - 2L

= 2L[(1+x^2/L^2)^(1/2) - 1] = 2L.x^2/(2L^2) = x^2/L

:. Longitudinal strain = (triangleL)/(2L) = x^2/(2L^2)

If T be the tension in the wire as shown figure then in equilibrium 2T cos theta = mg

or T = "mg"/(2costheta)

= "mg"/(2 x/(sqrt(x^2+L^2))) = (mgsqrt(x^2+L^2))/(2x) = "mgL"/(2x)

:. Stress = T/A = "mgL"/(2 x A)

As Young's modulus Y = "stress"/"strain"

=((mgL)/(2 x A))/((x^2)/(2L^2)) = (mgL)/(2 x A) xx (2L^2)/x^2 = (mgL^3)/(Ax^3)

=> x = [(mgL^3)/(YA)]^(1/3) = L[(mg)/(YA)]^(1/3)

= 1/2[(0.1xx9.8)/(2xx10^11xx0.50xx10^(-2)xx10^(-4))]^(1/3) = 1.074 xx 10^(-2) m

= 1.074 cm  ≈ 1.07 cm or 0.01 m

#### Solution 2

Length of the steel wire = 1.0 m

Area of cross-section, A = 0.50 × 10–2 cm= 0.50 × 10–6 m2

A mass 100 g is suspended from its midpoint.

m = 100 g = 0.1 kg

Hence, the wire dips, as shown in the given figure.

Original length = XZ

Depression = l

The length after mass m is attached to the wire = XO + OZ

Increase in the length of the wire:

Δ= (XO + OZ) – XZ

Where,

XO = OZ = [(0.5)^2 + l^2]^(1/2)

:.triangle l = 2[(0.5)^2 + (l)^2]^(1/2) - 1.0

= 2 xx 0.5 [1+(l/0.5)^2]^(1/2) - 1.0

Expanding and neglecting higher terms, we get

triangle l = l^2/0.5

"Strain" = "Increase in length"/"Original length"

Let T be the tension in the wire.

mg = 2T cosθ

Using the figure, it can be written as:

cos theta = l/((0.5)^2 + l^2)^(1/2)

= l/ ((0.5)(1+(l/0.5)^2)^(1/2))

Expanding the expression and eliminating the higher terms:

cos theta = l/((0.5)(1+l^2/(2(0.5)^2)))

(1+l^2/(0.5))=~1 for small l

:. cos theta =  l/(0.5)

:. T = (mg)/(2(l/(0.5))) = (mgxx 0.5)/(2l) = (mg)/(4l)

"Stress" = "Tension"/"Area" = (mg)/(4lxxA)

Y = (mg xx 0.5)/(4lxxAxxl^2)

l = sqrt((mgxx0.5)/(4YA))

Young’s modulus of steel, Y = 2 x 1011 Pa

:.l = sqrt((0.1xx9.8xx0.5)/(4xx2xx10^11xx0.50xx10^(-6)))

= 0.0106 m

Hence, the depression at the midpoint is 0.0106 m.

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#### APPEARS IN

NCERT Class 11 Physics Textbook
Chapter 9 Mechanical Properties of Solids
Q 19 | Page 245