# A force P is applied at a distance ‘X’ from the end of the beam as shown in the figure. What would be value of ‘X’ so that the displacement at ‘A’ is equal to zero?

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1. 0.5 L
2. 0.25 L
3. 0.33 L
4. 0.66 L

Option 4 : 0.66 L
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ISRO Scientist ME 2020 Paper
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## Detailed Solution

Concept:

FBD of diagram is

The net displacement should be zero.

So, displacement due to concentrated moment at B + displacement due to concentrated load P = 0

displacement due to concentrated moment at B $$= \frac{{P\left( {2L - x} \right) \cdot {{\left( {2L} \right)}^2}}}{{2EI}}$$ And

displacement due to concentrated load P $$=\frac{{P{{\left( {2L} \right)}^3}}}{{3EI}}$$

$$\Rightarrow \frac{{P\left( {2L - x} \right) \cdot {{\left( {2L} \right)}^2}}}{{2EI}} - \frac{{P{{\left( {2L} \right)}^3}}}{{3EI}}=0$$

$$\Rightarrow \frac{{P\left( {2L - x} \right) \cdot {{\left( {2L} \right)}^2}}}{{2EI}} = \frac{{P{{\left( {2L} \right)}^3}}}{{3EI}}$$

$$\frac{{2L - x}}{2} = \frac{{2L}}{3}$$

⇒ 6L – 3x = 4L    ⇒ 2L = 3x

$$\Rightarrow x = \frac{{2L}}{3} = 0.662$$
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