Volume of Submerged Solid: Solve Given Densities

[stratomire]

Homework Statement

Problem : An object is solid throughout. When the object is completely submerged in ethyl alcohol, its apparent weight is 15.2N. When completely immersed in water, its apparent weight is 13.7 N. What is the volume of the object? (The density of ethyl alcohol is 806 kg/m3 )

Homework Equations

SG = (W_A - W_L) / (W_A - W_W)

SG = ρL/ρW

ρ = m/v

W_A = mg

The Attempt at a Solution

I tried answering the problem by using the equation for specific gravity of a liquid lighter than water, which is:

SG = (W_A - W_L) / (W_A - W_W)

I know that SG can also be expressed as ρL/ρW, and from that I tried solving for V and got this equation:

V = (ρ_L W_A - ρ_W W_L) / [(ρ_L ρ_W) (g) (ρ)] ; g is 9.8 m/s2.

From this point on I was stuck.

I know the densities of water and ethyl alcohol because it was given but I don't know the density of the object so I just can't directly substitute the values. Because of this I am in great need of help, I can't think of another way to solve for the object's volume. Can you guys help me a little?

Thank you to those who'll be reading this and to those who'll try to help.

 

[alphysicist]

Hi stratomire,

Homework Statement

Problem : An object is solid throughout. When the object is completely submerged in ethyl alcohol, its apparent weight is 15.2N. When completely immersed in water, its apparent weight is 13.7 N. What is the volume of the object? (The density of ethyl alcohol is 806 kg/m3 )

Homework Equations

SG = (W_A - W_L) / (W_A - W_W)

I don't recognize this equation in the context of this problem. Isn't this related to the s.g. of a mixture?

Anyways, I don't think the specific gravity is the way to think approach this. Instead, when the problem says that the apparent weight in alcohol is 15.2N, how are the buoyant force and gravitational force related? What equation can you get from that?

Do the same for the apparent weight in water, and you should find two equations with two unknowns.

 

[thomate1]

Dear student,

Thank you for your question. It seems like you are on the right track with your approach using specific gravity. However, you are correct in identifying that you need the density of the object in order to solve for its volume.

To find the density of the object, we can use the equation ρ = m/v, where ρ is density, m is mass, and v is volume. We know the apparent weight (WA) of the object in both ethyl alcohol and water, so we can use the equation WA = mg to find the mass of the object (m). Once we have the mass, we can then use the given densities of ethyl alcohol and water to solve for the volume of the object.

Here is a step-by-step solution to the problem:

1. Use the equation WA = mg to find the mass of the object in ethyl alcohol and water.
WA = 15.2 N, WL = 13.7 N, g = 9.8 m/s^2
For ethyl alcohol:
15.2 N = m x 9.8 m/s^2
m = 1.55 kg
For water:
13.7 N = m x 9.8 m/s^2
m = 1.40 kg

2. Use the given densities to find the volume of the object in ethyl alcohol and water.
For ethyl alcohol:
ρ = 806 kg/m^3, m = 1.55 kg
V = m/ρ = 1.55 kg / 806 kg/m^3 = 0.00192 m^3
For water:
ρ = 1000 kg/m^3, m = 1.40 kg
V = m/ρ = 1.40 kg / 1000 kg/m^3 = 0.00140 m^3

3. Subtract the volume of the object in ethyl alcohol from the volume of the object in water to find the volume of the submerged part of the object.
Vsub = 0.00140 m^3 - 0.00192 m^3 = - 0.00052 m^3

4. Since the object is completely submerged, the volume of the submerged part is equal to the total volume of the object.
V = Vsub = - 0.00052 m^3

5. However, we cannot have a negative volume, so