Question

It is said that Archimedes discovered his principle during a bath while thinking about how he could determine if KingHiero‘s crown was actually made of pure gold. While in the bathtub, he conceived the idea that he could determine theaverage density of an irregularly shaped object by weighing it in air and also in water. If the crown weighed 3.20 kgf (=31.4 N) in air and 295 kgf (= 28.9 N) in water, determine if the crown is made of pure gold. The density of gold is 19,300kg/m3. Discuss how you can solve this problem without weighing the crown in water but by using an ordinary bucket withno calibration for volume. You may weigh anything in air.

Answer 1

Answer:

the crown is false densty= 12556kg/m^3[/tex]

Explanation:

Hello! The first step to solve this problem is to find the mass of the crown, this is found using the weight of the crown in the air by means of the equation for the weight.

W=mg

W=weight(N)=31.4N

M=Mass

g=gravity=9.81m/S^2

solving for M

m=W/g

$m=\frac{31.4N}{9.81m/S^2}=3.2kg$

The second step is find the volume of crown  remembering that when an object is weighed in the water the result is the subtraction between the weight of the object and the buoyant force of the water which is the product of the volume of the crown by gravity by density of water

$F=mg-\alpha V g$

Where

F=weight in water=28.9N

m=mass of crown=3.2kg

g=gravity=9.81m/S^2

α=density of water=1000kg/m^3

V= crown´s volume

solving for V

$V=\frac{mg-F }{g \alpha } =\frac{(3.2)(9.81)-28.9}{9.81(1000)} =0.000254m^3$

finally, we remember that the density is equal to the index between mass and volume

$\alpha =\frac{m}{v} =\frac{3.2}{0.000254} =12556kg/m^3$

To determine the density of the crown without using the weight in the water and with a bucket we can use the following steps.

1.weigh the crown in the air and find the mass

2. put water in a cylindrical bucket and measure its height with a ruler.

3. Put the crown in the bucket and measure the new water level with a ruler.

4. Subtract the heights, and find the volume of a cylinder knowing the difference in heights and the diameter of the bucket, in order to determine the volume of the crown.

5. find density by dividing mass by volume