JEE MAIN - Physics (2008 - No. 19)
While measuring the speed of sound by performing a resonance column experiment, a student gets the first resonance condition at a column length of $$18$$ $$cm$$ during winter. Repeating the same experiment during summer, she measures the column length to be $$x$$ $$cm$$ for the second resonance. Then
$$18 > x$$
$$x > 54$$
$$54 > x > 36$$
$$36 > x > 18$$
Explanation
For first resonant length $$v = {v \over {4{\ell _1}}} = {v \over {4 \times 18}}$$ (in winter)
For second resonant length
$$v' = {{3v'} \over {4{\ell _2}}} = {{3v'} \over {4x}}$$ (in summer)
$$\therefore$$ $${v \over {4 \times 18}} = {{3v'} \over {4 \times x}}$$
$$\therefore$$ $$x = 3 \times 18 \times {{v'} \over v}$$
$$\therefore$$ $$x = 54 \times {{v'} \over v}cm$$
$$v' > v$$ because velocity of light is greater in summer as compared to winter
$$\left( {v \propto \sqrt T } \right)$$
$$\therefore$$ $$x > 54\,cm$$
For second resonant length
$$v' = {{3v'} \over {4{\ell _2}}} = {{3v'} \over {4x}}$$ (in summer)
$$\therefore$$ $${v \over {4 \times 18}} = {{3v'} \over {4 \times x}}$$
$$\therefore$$ $$x = 3 \times 18 \times {{v'} \over v}$$
$$\therefore$$ $$x = 54 \times {{v'} \over v}cm$$
$$v' > v$$ because velocity of light is greater in summer as compared to winter
$$\left( {v \propto \sqrt T } \right)$$
$$\therefore$$ $$x > 54\,cm$$
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