In: Physics

Take the density of the crown to be \(\rho_{\mathrm{c}} .\) What is the ratio of the crown's apparent weight (in water) \(W_{\text {apparent }}\) to its actual weight \(W_{\text {actual }} ?\)

Express your answer in terms of the density of the crown \(\rho_{\mathrm{c}}\) and the density of water \(\rho_{\mathrm{w}}\)

\(\frac{W_{\text {apparent }}}{W_{\text {actual }}}=1-\frac{\rho_{w}}{\rho_{c}}\)

Concepts and reason

The concepts required to solve the given questions is apparent weight and the actual weight. Initially, calculate actual weight of the crown. Later, calculate the apparent weight of the crow. Finally, express the answer in terms of the density of crown and the density of water.

Fundamentals

The expression for the actual weight of the crown is as follows:

\(W=m g\)

\( =\rho_{\mathrm{c}} \)

Here, \(\rho_{\mathrm{c}}\) is the density of the crown, \(\mathrm{m}\) is the mass, and \(\mathrm{g}\) is the acceleration due to gravity. The expression for the apparent weight of the crown is as follows:

\(W_{\mathrm{a}}=W-F_{\mathrm{B}}\)

Here, \(F_{\mathrm{B}}\) is the force of buoyancy. The expression for the buoyance force is given by, \(F_{\mathrm{B}}=\rho_{\mathrm{w}} V g\)

Here, \(\rho_{\mathrm{w}}\) is the density of the water and \(V\) is the volume, and \(\mathrm{g}\) is the acceleration due to gravity.

Substitute \(\rho_{\mathrm{w}} V g\) for \(F_{\mathrm{B}}\) in the equation \(W_{\mathrm{a}}=W-F_{\mathrm{B}}\) \(W_{\mathrm{a}}=W-\rho_{\mathrm{w}} V g\)

Rearrange the above equation as follows:

\(\frac{W_{\mathrm{a}}}{W}=1-\frac{\rho_{\mathrm{W}} V g}{W}\)

The density is equal the mass divided by the volume. It is denoted by \(\rho\). The force of buoyancy is directly proportional to the density of the water, volume, and acceleration due to gravity.

Substitute \(\rho_{\mathrm{c}} V g\) for \(\mathrm{W}, \rho_{\mathrm{w}} V g\) for \(F_{\mathrm{B}}\) in the equation \(\frac{W_{\mathrm{a}}}{W}=1-\frac{\rho_{\mathrm{W}} V g}{W}\)

\(\frac{W \mathrm{a}}{W}=1-\frac{\rho_{\mathrm{W}} V g}{\rho \mathrm{c} V g}\)

\(=1-\frac{\rho_{\mathrm{W}}}{\rho_{\mathrm{c}}}\)

The ratio of the crown's apparent weight to its actual weight is equal to \(1-\frac{\rho_{\mathrm{W}}}{\rho_{\mathrm{c}}}\).

Buoyant force is the upward force exerted on the object placed in the fluid.

The ratio of the crown's apparent weight to its actual weight is equal to \(1-\frac{\rho_{\mathrm{W}}}{\rho \mathrm{c}}\).

It is said that Archimedes discovered the buoyancy laws when asked by King Hiero of Syracuse to determine whether his new crown was pure gold (SG = 19.3). Archimedes measured the weight of the crown in air to be 11.8 N and its weight in water to be 10.9 N. Was it pure gold?

Imagine that the apparent weight of the crown in water is \(W_{\text {apparent }}=4.50 \mathrm{~N},\) and the actua weight is \(W_{\text {actual }}=5.00 \mathrm{~N}\). Is the crown made of pure \((100 \%)\) gold? The density of water is \(\rho_{\mathrm{w}}=1.00\) grams per cubic centimeter. The density of gold is \(\rho_{\mathrm{g}}=19.32\) grams per cubic centimeter.
Yes
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