Finding the cavity volume in a gold sculpture from weight change in water

[monkeyman44]

I have a question regarding fluids

A small sculpture made of gold (density of gold = 19280 kg/m^3) is believed to have a secret central cavity. The weight of the sculpture in air is 25.76 N. When submerged in water, the weight is 23.86 N. What is the volume of the secret cavity. (Density of water = 1000 kg/m^3)

The answer is supposed to be volume = 5.7x(10^-5). I can't seem to get that answer. I believe my approach to the problem is wrong.

this is my approach but it is wrong


F=mg

25.76 N = m(9.8 m/s^2)
m = 2.6 kg

density = mass/volume

19280 kg/m^3 = 2.6 kg / v
v = 1.3 x 10^-4

i don't know what to do with the next part after... thanks

 

[Doc Al]

25.76 N = m(9.8 m/s^2)
m = 2.6 kg

This correct.

density = mass/volume

19280 kg/m^3 = 2.6 kg / v
v = 1.3 x 10^-4

This is the volume the sculpture would have if it were solid gold--with no cavity.

What is the buoyant force on the sculpture? (You can figure it out from the information given.) Use the buoyant force to calculate the actual volume of the sculpture. Then you can figure out the size of the cavity.

 

[wais]

First of all, great job on setting up the problem correctly! Your approach is on the right track, but there are a few things that need to be adjusted.

To find the volume of the secret cavity, we need to use the buoyant force equation:

Fb = ρVg

Where Fb is the buoyant force, ρ is the density of the fluid (in this case, water), V is the volume of the object submerged, and g is the acceleration due to gravity.

In this problem, the buoyant force is equal to the weight of the object in air minus the weight of the object in water:

Fb = 25.76 N - 23.86 N = 1.9 N

Using the buoyant force equation, we can solve for the volume of the object:

1.9 N = (1000 kg/m^3)V(9.8 m/s^2)

V = 1.9/(1000*9.8) = 1.93 x 10^-4 m^3

Since we know the total volume of the object (V = 1.3 x 10^-4 m^3) and the volume of the secret cavity (V = 1.93 x 10^-4 m^3), we can subtract to find the volume of the solid gold part of the sculpture:

V = 1.93 x 10^-4 m^3 - 1.3 x 10^-4 m^3 = 6.3 x 10^-5 m^3

Therefore, the volume of the secret cavity is:

Vsecret = 1.93 x 10^-4 m^3 - 6.3 x 10^-5 m^3 = 5.7 x 10^-5 m^3

I hope this helps clarify the problem and your approach! Keep up the good work.