A heavy plank of mass 100 kg hangs on three vertical wires of equal...

A heavy plank of mass $100$ kg hangs on three vertical wires of equal length arranged symmetrically (see figure). All the wires have the same cross-section. The middle wire is of steel and the other two are of copper. The modulus of elasticity of steel is assumed to be double that of copper. Determine the tensions in the wire. ( use $g = 9 . 8ms^{- 2}$ )

A heavy plank of mass 100 kg hangs on three vertical wires of equal length arranged symmetrically (see figure). All the

Options

A ${T}_{cu} = 245N$ , ${T}_{steel} = 490N$
B ${T}_{cu} = 235N$ , ${T}_{steel} = 490N$
C ${T}_{cu} = 205N$ , ${T}_{steel} = 490N$
D ${T}_{cu} = 295N$ , ${T}_{steel} = 420N$

Correct answer

A. ${T}_{cu} = 245N$ , ${T}_{steel} = 490N$

Step-by-step solution

Let tension in copper and steel wires are ${T}_{cu} and$ {T}_{st} $. Assuming same strain (due to heavy plank) we note that $ {Y}_{steel} = \dfrac{T{}_{st}L }{A $\left(\Delta L \right)}$ and

or $\dfrac{T{}_{st}}{T{}_{cu}} = \dfrac{Y{}_{steel}}{Y{}_{cu}} = 2 \left(\text{given}\right)or$ T{}_{st} = 2T{}_{cu} $ Further,$ {T}_{st} + 2{T}_{cu} = 100kg ( given ) $ <img alt="" src="/api/proxy-image?url=https%3A%2F%2Ffirebasestorage.googleapis.com%2Fv0%2Fb%2Fquantrexacademy-app.firebasestorage.app%2Fo%2Fquestions%252Ffigs%252Fquizrr%252Fquestion-assets%252F99_percentile_module%252Ffinal-question%252FEM0014785_AnsExp1_1.jpg%3Falt%3Dmedia \text{ and } clean=1 \text{ and } v=qxfig138" style="width: 310px; height: 216px;"/ data-qx-orig-src="/assets/diagrams/qx-book-b29534ec41080a2f.png" data-qx-storage-src="/assets/diagrams/qx-book-b29534ec41080a2f.png"> or$ 2{T}_{cu} + 2{T}_{cu} = 100kg $or$ {T}_{cu} = 25kg $ And therefore,$ {T}_{st} = 2{T}_{cu} = 50kg $ Thus$ {T}_{cu} $is$ {T}_{cu} = 25kgf = 245N $and$ {T}_{st} = 50kgf = 490N $$

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