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Elasticity Problems





In this page we have Elasticity Problems for JEE Main/Advanced And NEET . Hope you like them and do not forget to like , social share and comment at the end of the page.

Subjective Questions

Question 1
When a 100 Kg mass is suspended from wire of length 1 meter and and area 1 cm2, it length increases.
(a) Find the stress in the wire
(b) Find the increase in length of the wire
(c) Find the elastic potential energy stored in the wire
(d) Find the tension in the wire in loaded position

Answer

Given g=10 m/sec2 and Young Modulus of the material of the wire=2X1011 N/m2
Stress =Load/Area
=mgA=100×10104=107 N/m2
Now
Young Modulus=Stress/Strain
So strain= Stress/Young Modulus
=5×105
Now we know that
Strain = Increase in length/Initial length
=> Increase in Length=strain * Initial length
=5×105 m
Elastic Potential energy
=12stress×strain×Volume
=5×102 J
The tensions in the load position will the weight of the mass
So T=mg
=100*10=1000 N


Question 2
A thin metallic rod of Length L and radius of cross-section r rotates with the angular velocity in a horizontal plane about a vertical axis passing through one of its end.
Density of rod is ρ and Young Modulus of elasticity is Y. Breaking stress of the rod is S Find out the following
(a) Find the tension in the rod as a function of distance from the fixed end
(b) Find the tension at the fixed end, midpoint and end point of the rod
(c) Find the maximum angular velocity with which it can rotate with out breaking
(d) Find the total elongation in the rod

Answer

Let us consider a small element dx at a distance x from the rotating end.
Now mass of the rod =Lπr2ρ

Mass per unit length of the rod=πr2ρ
Mass of the small element=πr2dxρ
Centripetal force acting on the element
dF=πr2ρxω2dx
The Tension in the element would be due to the centripetal force of the outer portion i.e from x=x to x=L
F=Lxπr2ρxω2dx
F=12πr2ρω2(L2x2)
Tension at fixed point
x=0
F=ρπr2ω2L22
At x=L/2
F=3ρπr2ω2L8
So tension is maximum at fixed end and decrease towards outer end.
So
Sπr2>ρπr2ω2L22
Or 2SρL2<ω
Let dy be the elongation at the element of length dx
Then
dydx=1YF(x)A
Substituting the value of F(x) from previous part
dy=πr2ρω2(L2x2)dx2Yπr2
y=L0ρω22Y(L2x2)dx=ρω2L33Y



Question 3
An elastic string has a mass M suspended at its lower end,the upper being fixed to a support. When the mass is pulled down over a short distance and let go, then it executes SHM with Time period given by
T=2πl1g
Where l1 is the elongation of the string.Now the mass m is added to the mass M, then it is found that Time period of oscillation (T2) such that
T1T2=54
Find the ratio m:M
Given Y is the Young modulus of elasticity. L be the original length of string.

Answer

Let l2 be the elongation in second case then
T1=2πl1g ---(1)
T2=2πl2g ---(2)

So that
T21T22=l1l2
Or
l1l2=1625
Now
For first case

Y=MgLAl1 --(3)
Second case
Y=(M+m)gLAl2 --(4)
From 3 and 4
l1l2=MM+m
Or
MM+m=1625

M+mM=2516
Or
M+mMM=251619
Or
mM=916


Question 4
An equilateral triangle ABC is formed by two Cu rods AB and BC and one Al rod. It is heated in such a way that temperature of each rod increases by ΔT. Find change in the angle ABC. [Coeff. of linear expansion for Cu is α1 ,Coeff. of linear expansion for Al is alpha2 ]

Answer

Let l1 = AB, l2 = AC, l3 = BC
cosθ=l23+l21l222l3L1
or
2l3L1cosθ=l23+l21l22
Differentiating
2(l3dL1+l1dl3)cosθ2l3L1sinθdθ=2l3dl3+2l1dl12l2dl2
Now
dl1=l1α1ΔT
dl2=l2α1ΔT
dl3=l3α2ΔT
and l1=l2=l3=l
Substituting these values
2(l2α1ΔT+l2α2ΔT)cosθ2l2sinθdθ=2l2α1ΔT+2l2α1ΔT2l2α2ΔT
sinθdθ=2α1ΔT(1cosθ)α2ΔT
Substituting θ=600
dθ=2(α1α2)ΔT3



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