Buoyancy and Archimedes' Principle

[Soaring Crane]

If an object floats in water, its density can be determined by tying a sinker on it so that both the object and sinker are submerged. Show that the specific gravity, (Density_substance)/(density_water at 4 degrees), is given by (w)/(w_1 - w_2), where w is the object's weight in air alone, w_1 is the apparent weight when a sinker is tied to it , and the sinker is submerged, and w_2 is the apparent weight when both the object and sinker are submerged.

Could anyone give me some pointers on what to do?

Thanks.

 

[Curious3141]

Archimedes' principle : buoyant force on an object is equal to the weight of the water displaced. The weight of the water displaced is of course equal to the volume of the object being immersed multiplied by the density of water.

For this problem, you're expected to disregard the buoyancy due to air, that is, treat the weights measured in air as "true" weights.

Define : $w_s, v_o, v_s, \rho_w, \rho_o, m_o, g$ as true weight of sinker, volume of object, volume of sinker, density of water, density of object, mass of object and gravitational acceleration respectively. The other definitions are as given in the problem statement.

Set up 3 equations like so :

$$w = (m_o)(g) = (v_o)(\rho_o)(g)$$ ---(1)

$$w_1 = (w + w_s) - (v_s)(\rho_w)(g)$$ ---(2)

$$w_2 = (w + w_s) - (v_o + v_s)(\rho_w)(g)$$ ---(3)

Take (2) - (3) :

$$w_1 - w_2 = (v_o)(\rho_w)(g)$$ --- (4)

Take (1)/(4) :

$$\frac{w}{w_1 - w_2} = \frac{\rho_o}{\rho_w}$$

And you're done. If you need further explanation please post.

 

[wais]

Sure! Buoyancy and Archimedes' Principle state that the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. This principle can be used to determine the density of an object by using a sinker and measuring its apparent weight in different situations.

First, let's define some variables:
- w: weight of the object in air alone
- w1: apparent weight of the object when a sinker is tied to it and both are submerged
- w2: apparent weight of both the object and sinker when submerged
- ρ_substance: density of the object
- ρ_water: density of water at 4 degrees (we will assume this is constant)

Using Archimedes' Principle, we can set up the following equations:
w = ρ_substance * V * g (where V is the volume of the object and g is the acceleration due to gravity)
w1 = (ρ_substance + ρ_sinker) * V * g (since the object and sinker are submerged together)
w2 = ρ_water * V * g (since the object and sinker displace a volume of water equal to their combined volume)

Now, let's rearrange these equations to solve for ρ_substance and ρ_sinker:
ρ_substance = w / (V * g)
ρ_sinker = (w1 - w2) / (V * g)

Since we are interested in the specific gravity, we can divide both equations by ρ_water * V * g:
ρ_substance / (ρ_water * V * g) = w / (w2 - w1)
ρ_substance / ρ_water = w / (w2 - w1)

Therefore, the specific gravity is given by:
ρ_substance / ρ_water = w / (w2 - w1)

I hope this helps! Let me know if you have any other questions.