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A rod of length 2m and cross sectional area 8x 10–6m2 is clamped between two rigid supports. If the wire is heated through, 25°C, then find the thermal stress and compressional force set up in the wire. The Young's modulus of rod is 2x1011N/m2 and its coefficient of linear expansion is 1.6 x 10–5 per°C. 


The thermal stress set up in the wire is


The thermal stress set up in the wire is
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Two wires P and Q of same length and material but radii in the ratio 2:1 are suspended from a rigid support. Find the ratio of strain produced in the wires when: (i) both are under same stress (ii) both are loaded by same weight.

Given the radii of two wires is of the ratio 2:1.
Using the formula as per Hooke's law, 

Strain  Stress =LoadArea=Fπr2and           rP : rQ=2 : 1

i) When both the wires are under the same stress, strain produced will be the same.

ii) When both the wires are loaded by the same weight, then

StrainpStrainQ=(rQ)2(rP)2=14 




 

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A wire increases by 10–3 of its length, when a stress of 108N/m2 is applied on it. What is the Young's modulus of material of wire?

Given,
Wire is increased of it's length, i.e., strain = 10-3 
Stress = 108N/m2

 Young's modulus = StressStrain= 10810-3= 1011 Nm-2  

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When a wire is loaded by weight M1, its length is Land when loaded by weight Mits length becomes L2. Find the original length of the wire. 

L1 wire is loaded by weight Mand when M2 is loaded length becomes M2

Let L be the original length, A be the area of cross section of wire and Y be Young's modulus of the material of wire.
Therefore, the extension in wire is (L
1–L) when loaded by weight M1 and that is (L– L) when loaded by weight M2.

Thus, Young's modulus is,

Y = M1g×LA(L1 - L)= M2g×LA(L2 - L)        M1(L1-L) = M2(L2 - L) M1L2 - M1L1 = M2L1 -M1L2                      L = M2L1 - M1L2(M2-M1)

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A wire loaded by weight of density 7800 kg/m3 is stretched to length of 1m. On immersing the weight in liquid of density 1300kg/m3, the length shortens by 6mm. Find the original length of the wire.


Given,

Density of the load, ρ = 7800 kg/m3 
Length of the wire, l = 1 m
Density of liquid, = 1300 kg/m3

Let, V be the volume of weight suspended from the wire.
 
Therefore, tension in the wire when load is in the air is equal to, 
 

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