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Young’s modulus (Y)

Young’s modulus is defined as the ratio of normal stress to longitudinal strain within the limit of proportionality.
 
48531.png
 
If force is applied on a wire of radius r by hanging a weight of mass M, then
 
48525.png
  
Important Points
  • If the length of a wire is doubled, then
     
    Longitudinal strain = 48975.png
     
    48969.png
     
    ∴ Young’s modulus = 48963.png
     
    ⇒ Y = Stress[as strain = 1]
     
    So Young’s modulus is numerically equal to the stress which will double the length of a wire.
  • Increment in the length of wire:
     
    50154.png50150.png
     
    So, if same stretching force is applied to different wires of same material, 50146.png[as F and Y are constant.]
     
    i.e., greater the ratio L r2, greater will be the elongation in the wire.
 
Elongation in a wire by its own weight The weight of the wire Mg acts at the center of gravity of the wire so that the length of wire which is stretched will be L / 2.
 
∴ Elongation, 48494.png = 48488.png
[as mass (M) = volume (AL) × density (d)]
 
Thermal stress If a rod is fixed between two rigid supports, due to change in temperature its length will change and so it will exert a normal stress (compressive if temperature increases and tensile if temperature decreases) on the supports. This stress is called thermal stress (Fig. 11).
 
48482.png
Fig. 11
 
As by definition, coefficient of linear expansion, 48476.png
 
Thermal strain, 48470.png  
So thermal stress = Yα Δθ [as Y = stress/strain]
 
and tensile or compressive force produced in the body = YAα Δθ

 

NoteIn case of volume expansion, thermal stress = Kγ Δθ, where K = Bulk modulus and γ = coefficient of cubical expansion.

 

Force between two rods Two rods of different metals having the same area of cross sectionA are placed end to end between two massive walls as shown in Fig. 12.
 
50289.png
Fig. 12
 
The first rod has a length L1, coefficient of linear expansion α1, and Young’s modulus Y1. The corresponding quantities for second rod are L2α2, and Y2. If the temperature of both the rods is now raised by T degrees then increase in length of the composite rod (due to heating) will be equal to
 
l1 + l2 = [L1α1 + L2α2]T [as l = L α Δθ]
 
and due to compressive force F from the walls due to elasticity,
 
decrease in length of the composite rod = 48432.png
48426.png
 
As the length of the composite rod remains unchanged, increase in length due to heating must be equal to decrease in length due to compression, i.e.
 
48420.png
or 48414.png 
 
Force constant of wire Force required to produce unit elongation in a wire is called force constant of material of wire. It is denoted by k.
 
But from the definition of young’s modulus, 48402.png.
 
 
From (1) and (2),
 
48389.png
 
It is clear that the value of force constant depends upon the dimension (length and area of cross section) and material of a substance.
 
Actual length of the wire If the actual length of the wire is L, then under tension T1, its length becomes L1 and under tension T2, its length becomes L2.
 
 
From (3) and (4), we get
 
48371.png




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