Motion of an oscillating liquid column in a U-tube is

(a) Periodic but not simple harmonic.

(b) Non-periodic.

(c) Simple harmonic and time period is independent of the density of the liquid.

(d) Simple harmonic and time-period is directly proportional to the density of the liquid.

<p><span data-teams="true">This is a multiple choice answer as classified in NCERT Exemplar</span></p><p> (c) Consider the diagram in which a liquid column oscillates . in this case, restoring force acts on the liquid due to gravity.</p><p>Restoring force f = weight of liquid column of height 2y</p><p>t=-A <span contenteditable="false"> <math> <mo>×</mo> <mn>2</mn> <mi>y</mi> <mo>×</mo> <mi>ρ</mi> <mo>×</mo> <mi>g</mi> </math> </span>= -2A<span contenteditable="false"> <math> <mi>ρ</mi> <mi>g</mi> <mi>y</mi> </math> </span></p><p>f<span contenteditable="false"> <math> <mo>∝</mo> <mo>-</mo> <mi>y</mi> </math> </span></p><p>motion is SHM with force constant k= 2A<span contenteditable="false"> <math> <mi>ρ</mi> <mi>g</mi> </math> </span></p><p>T=<span contenteditable="false"> <math> <mn>2</mn> <mi>π</mi> <mroot> <mrow> <mrow> <mfrac> <mrow> <mrow> <mi>m</mi> </mrow> </mrow> <mrow> <mrow> <mi>k</mi> </mrow> </mrow> </mfrac> </mrow> </mrow> <mrow></mrow> </mroot> </math> </span> = 2<span contenteditable="false"> <math> <mi>π</mi> <mroot> <mrow> <mrow> <mfrac> <mrow> <mrow> <mi>A</mi> <mo>×</mo> <mn>2</mn> <mi>h</mi> <mo>×</mo> <mi>ρ</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> <mi>A</mi> <mi>ρ</mi> <mi>g</mi> </mrow> </mrow> </mfrac> </mrow> </mrow> <mrow></mrow> </mroot> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mroot> <mrow> <mrow> <mfrac> <mrow> <mrow> <mi>h</mi> </mrow> </mrow> <mrow> <mrow> <mi>g</mi> </mrow> </mrow> </mfrac> </mrow> </mrow> <mrow></mrow> </mroot> </math> </span></p><p>So time period is independent upon density of liquid.</p>

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