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Chapter 7: Problem 1031

The fraction of floating object of volume \(\mathrm{V}_{0}\) and density \(\mathrm{d}_{0}\) above the surface of a Liquid as density \(\mathrm{d}\) will be (A) \(\left(\mathrm{d}_{0} / \mathrm{d}\right)\) (B) $\left[\left\\{\mathrm{dd}_{0}\right\\} /\left\\{\mathrm{d}+\mathrm{d}_{0}\right\\}\right]$ (C) \(\left[\left\\{d-d_{0}\right\\} / d\right]\) (D) $\left[\left\\{\mathrm{dd}_{0}\right\\} /\left\\{\mathrm{d}-\mathrm{d}_{0}\right\\}\right]$

Short Answer

Expert verified
The short answer is: \(\left[\left\\{d-d_{0}\right\\} / d\right]\).

Step by step solution

01

Find the weight of the object

We can find the weight of the object (W_object) by multiplying its volume (V0) by its density (d0) and gravitational acceleration (g). So, \(W_\text{object} = V_0 d_0 g \).
02

Find the weight of the liquid displaced

The weight of the liquid displaced (W_liquid) can be found by the volume of the object submerged in the liquid (V_submerged) times the density of the liquid (d) and gravitational acceleration (g). Since we want the fraction of the object above the surface, we find the fraction submerged f_submerged, which is equal to the object's density (d0) divided by the liquid's density (d) i.e., \(f_\text{submerged} = \dfrac{d_0}{d}\). Now, the volume submerged (V_submerged) is the fraction submerged (f_submerged) times the total volume (V0) i.e., \(V_\text{submerged} = V_0 f_\text{submerged}\). So, \(W_\text{liquid} = V_\text{submerged} d g = V_0 f_\text{submerged} d g\).
03

Find the fraction of the object above the surface

According to Archimedes' principle, the weight of the liquid displaced (W_liquid) is equal to the weight of the object (W_object). So, \(V_0 d_0 g = V_0 f_\text{submerged} d g\). We can cancel out V0 and g from this equation, leaving us with \(d_0 = f_\text{submerged} d\). Now, we can solve for the fraction above the surface (f_above) by noting that it's equal to 1 minus the fraction submerged (f_submerged), so \(f_\text{above} = 1 - \dfrac{d_0}{d}\). Comparing the expression for the fraction above the surface with the given options, we find that the correct answer is: (C) \(\left[\left\\{d-d_{0}\right\\} / d\right]\)

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