A cubical block of density ρ is floating on the surface of water. Out of its height L, fraction x is submerged in water. The vessel is in an elevator accelerating upward with acceleration a . What is the fraction immersed?

This is a short answer type question as classified in NCERT Exemplar

Consider the diagram.

Let the density of water be ${\rho }_w$  and a cubical block of ice of side L be floating in water with x of its height L submerged in water.

Volume of the block V = L³

Mass of the block m = V $\rho$ =L^{3 $\rho$}

Weight of the block = mg= L^{3 $\rho g$}

1st case

Volume of the water displaced by the submerged part of the block= xL²

Weight of the water displaced by the block

In floating condition, x L²

Weight of the block= Weight of the water displaced by the block

L^{3 $\rho g$} = xL^{2 ${\rho }_{w}g$}

$\frac{x}{L}=\frac{\rho }{{\rho }_w}=x$

2nd case

When elevator is accelerating upward with an acceleration a, then effective acceleration

= (g+a)

Then, w

...more

This is a short answer type question as classified in NCERT Exemplar

Consider the diagram.

Let the density of water be ${\rho }_w$  and a cubical block of ice of side L be floating in water with x of its height L submerged in water.

Volume of the block V = L³

Mass of the block m = V $\rho$ =L^{3 $\rho$}

Weight of the block = mg= L^{3 $\rho g$}

1st case

Volume of the water displaced by the submerged part of the block= xL²

Weight of the water displaced by the block

In floating condition, x L²

Weight of the block= Weight of the water displaced by the block

L^{3 $\rho g$} = xL^{2 ${\rho }_{w}g$}

$\frac{x}{L}=\frac{\rho }{{\rho }_w}=x$

2nd case

When elevator is accelerating upward with an acceleration a, then effective acceleration

= (g+a)

Then, weight of the block =m (g+a)

= L^{3 $\rho$} (g+a)

Let x₁ fraction be submerged in water when elevator is accelerating upwards.

Now, in the floating condition, weight of the block= weight of the displaced water

L^{3 $\rho (g+a$} )= ( $x_{1}L$ ²) ${\rho }_w$ (g+a)

$\frac{x_1}{L}=\frac{\rho }{{\rho }_w}$  = x

From1^st and 2^nd case,

We see that, the fraction of the block submerged in water is independent of the acceleration of the elevator.

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<p><span data-teams="true">This is a short answer type question as classified in NCERT Exemplar</span></p><p>Consider the diagram.</p><p>Let the density of water be <span contenteditable="false"> <math> <msub> <mrow> <mrow> <mi>ρ</mi> </mrow> </mrow> <mrow> <mrow> <mi>w</mi> </mrow> </mrow> </msub> </math> </span>&nbsp;and a cubical block of ice of side L be floating in water with x of its height L submerged in water.</p><p>Volume of the block V = L³</p><p>Mass of the block m = V<span contenteditable="false"> <math> <mi>ρ</mi> </math> </span>=L^{3<span contenteditable="false"> <math> <mi>ρ</mi> </math> </span>}</p><div><div><picture><source srcset="https://images.shiksha.com/mediadata/images/articles/1745474256phpzj1EaK_480x360.jpeg" media=" (max-width: 500px)"><img src="https://images.shiksha.com/mediadata/images/articles/1745474256phpzj1EaK.jpeg" alt="" width="142" height="125"></picture></div></div><p>Weight of the block = mg= L^{3<span contenteditable="false"> <math> <mi>ρ</mi> <mi>g</mi> </math> </span>}</p><p><strong>1st case</strong></p><p>Volume of the water displaced by the submerged part of the block= xL²</p><p>Weight of the water displaced by the block</p><p>In floating condition, x L²</p><p>Weight of the block= Weight of the water displaced by the block</p><p>L^{3<span contenteditable="false"> <math> <mi>ρ</mi> <mi>g</mi> </math> </span>}= xL^{2<span contenteditable="false"> <math> <msub> <mrow> <mrow> <mi>ρ</mi> </mrow> </mrow> <mrow> <mrow> <mi>w</mi> </mrow> </mrow> </msub> <mi>g</mi> </math> </span>}</p><p><span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <mi>x</mi> </mrow> </mrow> <mrow> <mrow> <mi>L</mi> </mrow> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mrow> <mi>ρ</mi> </mrow> </mrow> <mrow> <mrow> <msub> <mrow> <mrow> <mi>ρ</mi> </mrow> </mrow> <mrow> <mrow> <mi>w</mi> </mrow> </mrow> </msub> </mrow> </mrow> </mfrac> <mo>=</mo> <mi>x</mi> </math> </span></p><p><strong>2nd case</strong></p><p>When elevator is accelerating upward with an acceleration a, then effective acceleration</p><p>= (g+a)</p><p>Then, weight of the block =m (g+a)</p><p>= L^{3<span contenteditable="false"> <math> <mi>ρ</mi> </math> </span>} (g+a)</p><p>Let x₁ fraction be submerged in water when elevator is accelerating upwards.</p><p>Now, in the floating condition, weight of the block= weight of the displaced water</p><p>L^{3<span contenteditable="false"> <math> <mi>ρ</mi> <mo> (</mo> <mi>g</mi> <mo>+</mo> <mi>a</mi> </math> </span>})= (<span contenteditable="false"> <math> <msub> <mrow> <mrow> <mi>x</mi> </mrow> </mrow> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> </msub> <mi>L</mi> </math> </span>²)<span contenteditable="false"> <math> <msub> <mrow> <mrow> <mi>ρ</mi> </mrow> </mrow> <mrow> <mrow> <mi>w</mi> </mrow> </mrow> </msub> </math> </span> (g+a)</p><p><span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <msub> <mrow> <mrow> <mi>x</mi> </mrow> </mrow> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> </msub> </mrow> </mrow> <mrow> <mrow> <mi>L</mi> </mrow> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mrow> <mi>ρ</mi> </mrow> </mrow> <mrow> <mrow> <msub> <mrow> <mrow> <mi>ρ</mi> </mrow> </mrow> <mrow> <mrow> <mi>w</mi> </mrow> </mrow> </msub> </mrow> </mrow> </mfrac> </math> </span>&nbsp;= x</p><p>From1^st and 2^nd case, </p><p>We see that, the fraction of the block submerged in water is independent of the acceleration of the elevator.</p>

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