Question

The length of a rubber cord is \[{l_1}\]meters when the tension in it is \[4N\] and \[{l_2}\]​meters when the tension is \[5N\]. then the length in meters when the tension is \[9N\] is
(A) \[3{l_2} + 4{l_1}\]
(B) \[3{l_2} + 2{l_1}\]
(C) \[5{l_2} - 4{l_1}\]
(D) \[3{l_2} - 2{l_1}\]

Answer 1

Hint Let us assume that the original length of the rubber cord to be l meters. When a force of\[4N\]is applied on the cord it changes to \[{l_1}\]and when \[5N\]is applied it changes to \[{l_2}\]. Now using Young's modulus formula, we can relate the tension and change in length from original to find the change in length when extra force is applied.

Complete Step By Step Answer
 It is given that a rubber cord , which originally has a length of \[l\]meters, undergoes two different tensions of variable magnitude and undergoes expansion or change in length. Whenever a material undergoes a tension of particular magnitude, we consider it’s tensile property to analyze whether the material is stiff or undergoes elasticity.
To determine the stiffness of any component, we use Young's modulus of elasticity to determine whether the given material will remain stiff or not under applied forces or tension. Mathematically, it can be represented as ratio between stress and strain and also as :
\[E = \dfrac{{TL}}{{A\Delta L}}\], where T is the tension experienced, L is the length of the cord, A is the area of the cord
Applying this for the first condition where the tension experienced is \[4N\]we get,
\[ \Rightarrow E = \dfrac{{4l}}{{A({l_1} - l)}}\]
Applying this for the second and third conditions respectively , we get
\[ \Rightarrow E = \dfrac{{5l}}{{A({l_2} - l)}}\] and \[ \Rightarrow E = \dfrac{{9l}}{{A({l_3} - l)}}\]
Equating all the E values, we get
\[ \Rightarrow E = \dfrac{{4l}}{{A({l_1} - l)}} = \dfrac{{5l}}{{A({l_2} - l)}} = \dfrac{{9l}}{{A({l_3} - l)}}\]
Equate 1 and 3 initially and 2 and 3 separately, so as to find \[{l_3}\],this implies
\[ \Rightarrow E = \dfrac{{4l}}{{A({l_1} - l)}} = \dfrac{{9l}}{{A({l_3} - l)}}\]
\[ \Rightarrow 4{l_3} + 5l = 9{l_1}\]-----(1)
Equating 2 and 3 we get
\[ \Rightarrow E = \dfrac{{5l}}{{A({l_2} - l)}} = \dfrac{{9l}}{{A({l_3} - l)}}\]
\[ \Rightarrow 5{l_3} + 4l = 9{l_2}\]--(2)
On solving 1 and 2 , we get
\[25{l_3} + 20l = 45{l_2}[(2) \times 5]\]
\[16{l_3} + 20l = 36{l_1}[(1) \times 4]\]
Cancelling out common 20l term, we get
\[ \Rightarrow {l_3} = 5{l_2} - 4{l_1}\]
Thus the length of the cord ,when a tension of 9N is applied is \[5{l_2} - 4{l_1}\]

Hence, Option (c) is the right answer for the given question.

Note In general, elasticity of a material is defined as the property of the material to resist distortion due to constant application of tension , undergo deformation and return back to its original shape and size once the force applied is removed.