Are the units of volume compatible with the density units?

[CMATT]

<post removed from here by mentor, but copied to a later post>

 

[nick gear]

Homework Statement

So, just took an exam, one of the questions on the exam was the following:

An object is attached to a scale and submerged in water (p=1000 kg/m^3) and it's weight reads 34N. The same object is then submerged in oil (p=830 kg/m^3) and it's weight reads 34.7N.

a.) What is the objects volume?
b.) What is the objects density?

What I know: (p= m/v) (m=weight/g) (V=Fbuoyancy/ p(g)) (Fbuoyancy= p(v))

For the life of me I couldn't answer this question, could someone please help, I will get the chance to make corrections.

Homework Equations

What I know: (p= m/v) (m=weight/g) (V=Fbuoyancy/ p(g)) (Fbuoyancy= p(v))

The Attempt at a Solution

T+Fb-mg=0
T1+(p*V1*g)-mg=0
V= T1+(p*g)-mg

This is the clue my proffesor gave us prior to exam. However when I plugged in numbers it seemed way off.
I know how to derive density with weight in air also known, but I don't have a clue where to begin with so little data.

 

[haruspex]

Start with Archimedes' principle. If the object has volume V, what weight of oil will it displace? What weight of water?

 

[SteamKing]

Homework Statement

So, just took an exam, one of the questions on the exam was the following:

An object is attached to a scale and submerged in water (p=1000 kg/m^3) and it's weight reads 34N. The same object is then submerged in oil (p=830 kg/m^3) and it's weight reads 34.7N.

a.) What is the objects volume?
b.) What is the objects density?

What I know: (p= m/v) (m=weight/g) (V=Fbuoyancy/ p(g)) (Fbuoyancy= p(v))

For the life of me I couldn't answer this question, could someone please help, I will get the chance to make corrections.

Homework Equations

What I know: (p= m/v) (m=weight/g) (V=Fbuoyancy/ p(g)) (Fbuoyancy= p(v))

The Attempt at a Solution

T+Fb-mg=0
T1+(p*V1*g)-mg=0
V= T1+(p*g)-mg

This is the clue my proffesor gave us prior to exam. However when I plugged in numbers it seemed way off.
I know how to derive density with weight in air also known, but I don't have a clue where to begin with so little data.

Fb = ρgV

where

Fb = buoyant force, N
ρ = density of the fluid, kg/m³
g = acceleration due to gravity, m/s²
V = submerged volume of the body, m³

You should always check to make sure your formulas give consistent units.

 

[Chandra Prayaga]

To me it seems odd that the weight is in N, rather than grams, or kilograms. Also are you trying to find the objects volume and density while it is in oil or while it is in water? Cause depending on that, you will get a different answer
Not sure if that really helped at all

No. Weight is a force and its unit IS a Newton. Grams and kilograms are units of MASS, not weight. So the statement of the problem is correct. Also, the density and volume are properties of the sample, dependent on temperature, not on what you dip the sample in.

 

[Chandra Prayaga]

Don't think of it as a formula into which you somehow plug in numbers. Think of the process and the principle that governs the process. If you submerge an object in a liquid, the principle of Archimedes has something to say about what happens. Start from there. State the principle, then use the data in the principle to find what you need.

 

[nick gear]

Archimedes' principle indicates that the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid that the body displaces.

In this case the object IS fully submerged, however I don't have a weight of fluid dispersed. I do however have weight while in water, and weight while in oil with a difference of .7N. I can find mass while in the water and oil ( m=W/g ) however I can't seem to figure out volume without Fbuoyancy (Fb= mg = (p*V*g)).

 

[nick gear]

It definitely seems my professor goes out of his way to derive problems that google or youtube can't find. lol

I've found every other way to approach density problems except this way.

 

[nick gear]

The textbook doesn't offer anything about this approach either.

 

[fresh_42]

Take the volume of the sample as variable V, W its weight in air. Then you know the weight of water it displaces depending on V. The same with the oil. So you get two values of reduced weight and you already know the difference: 0.7N. All depend on V. 2 equations, 2 variables. Density follows from V and W plus g.

 

[nick gear]

I gotcha, I do remember my professor saying something along the lines of comparison of the separate equations. Let me give that a shot, I'll get back at ya.

 

[nick gear]

Ok so this is what I got for just the submerged in water part. I know that if I can find volume of the object I can find density (d = m/V )

V= T + (p*g) - mg

wouldn't tension be equal to mg and cancel, therefor leaving (p*g) this number would seem to large though

V= (1000 kg/m^3 * 9.8 m/s^2)
V= 9800

 

[nick gear]

forgot units: V = 9800 m^3

 

[nick gear]

Archimedes' principle indicates that the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid that the body displaces.

The object obviously has volume and has a weight of 34N (water) and 34.7N (oil). To answer the second and third question, what weight in oil will it displace? The object will displace the same volume as the water it displaced, however how do I determine weight of displaced fluid when all I know is weight of the object while submerged?

 

[fresh_42]

To be honest, that confuses me. Why is there tension? What is p*g? Do the units fit together? If p should be a density, then you'll have $m^3$ as numerator on the left side and as denominator on the right.

In the first step you could calculate with the differences. Let W denote the weight of your sample, V its volume, m its mass, g Earth's acceleration and d its density.
You know the differences in density and the difference in weight, so $0.7 \cdot N = 170 \cdot kg \cdot m^{-3} \cdot g \cdot V$. From there on you can go ahead with V to calculate weight (using the water measurement of weight, weight of displaced water + 34N), mass and density.
I really recommend you to carry all units explicitly within your equations. Thus you know whether it comes out right!

 

[nick gear]

• So... V=170 *kg *m^-3 * (9.8 m/s^2) all divided by .7

 

[SteamKing]

Archimedes' principle indicates that the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid that the body displaces.

The object obviously has volume and has a weight of 34N (water) and 34.7N (oil). To answer the second and third question, what weight in oil will it displace? The object will displace the same volume as the water it displaced, however how do I determine weight of displaced fluid when all I know is weight of the object while submerged?

You know two important pieces of information: the mass of the object and its volume do not change when the object is submerged in oil or in water.

You can write an equation to determine the mass of fluid displaced by using the volume of the object and the density of the fluid. In fact, you already had a good equation, but you wrote it in such a manner to obscure its usefulness:

T1+(p*V1*g)-mg=0

It's better to write this:

Wfluid = ρgV - mg, where Wfluid is the weight of the object when it is submerged in a fluid having a density ρ. V is the volume of the object.

Once you write this equation for one fluid, you can write it for the other fluid. The difference of the weight of the object in the two fluids depends only on the difference in the density of the fluids, since the volume of the object doesn't change.

Once you figure out the volume of the object, then you can use Archimedes' Principle to find the mass of the object. Your Uncle Bob then drops by for a visit, and knowing both the mass of the object and its volume, he leaves you with the density.

 

[nick gear]

... all divided by .7 * N

 

[nick gear]

= 2380 m^3

 

[fresh_42]

... all divided by .7 * N

No! The other way round. But with N it's better and since $N = kg \cdot m \cdot s^{-2}$ only $m^3$ will be left. (Hint: it's a little lighter than lead and heavier than iron; if I made no mistake.)

 

[nick gear]

ohhhhhh! ok let me try that

 

[fresh_42]

The density d we are looking for is actually the following:
$d = \frac{34.7 N * 1000 kg*m^{-3} - 34 N * 830 kg*m^{-3} }{0.7 N}$

EDIT: I apologize. It's far too late here. Just thought I messed up the fluids and their densities.

 

[nick gear]

So...

V1= ((1000 (kg/m^3)(9.8 (m/s^2)) - 34N) / (34N) = 287.24 m^3
&
V2= ((830 (kg/m^3)(9.8 (m/s^2)) - 34.7N) / (34.7N) = 234.41 m^3

Vfinal = 287.24 m^3 - 234.41 m^3 = 52.83 m^3

density = (m1 - m2) / Vfinal = (3.47 (kg) - 3.54 (kg)) / (52.83 m^3) = (.07N) / (52.83 m^3) = 1.32 x 5^-10

 

[haruspex]

(1000 (kg/m^3)(9.8 (m/s^2)) - 34N)

That's dimensionally inconsistent. The first term is N/m³, the second is N.

 

[fresh_42]

... and stop switching threads with the same question!

 

[nick gear]

Sorry about it wasn't until after my first thread that I realized that the rules say to post on homework. lol

 

[nick gear]

so...

(34700 - 28220) / (.7) = 9.257 g/cm^3 which is close to Polonium @ 9.3 g/cm^3

 

[fresh_42]

I really hope it wasn't! Now you can calculate mass, weight and volume, too. And it'll fit into an aquarium, no more thousands of $m^3$ please!

EDIT: By the way, I recognized afterwards that it is well explained on the German Wiki-page for density, including exactly your problem! https://de.wikipedia.org/wiki/Dichte#Dichtebestimmung_durch_Auftrieb

 

[Mark44]

Copied from a post that was out of sequence due to my merging of two separate threads.

To me it seems odd that the weight is in N, rather than grams, or kilograms. Also are you trying to find the objects volume and density while it is in oil or while it is in water? Cause depending on that, you will get a different answer
Not sure if that really helped at all

 

[nick gear]

so here goes... I know its been a while, been wrapping up this semester. But this is what I came up with.

V= (34.7 N - 34 N) / (9.8 m/s^2) * (1000 - 803)
V= 3.625 x 10^ - 4 m^3 = .3681 L = 368.1 cm^3

 

[nick gear]

sorry 362.5 cm^3 which would fit into a .0958 gal aquarium

 

[nick gear]

now I need to find density
p = m/v What mass should I use? the weight of the scale for water, oil, or either?

 

[SteamKing]

so here goes... I know its been a while, been wrapping up this semester. But this is what I came up with.

V= (34.7 N - 34 N) / (9.8 m/s^2) * (1000 - 803)
V= 3.625 x 10^ - 4 m^3 = .3681 L = 368.1 cm^3

You used 803 kg/m³ as the density of the oil instead of the correct density of 830 kg/m³. Your calculated volume is wrong.

It's important to check your work for silly copying mistakes. It saves hours of pulling your hair out, wondering why you don't get the right answer.

 

[SteamKing]

now I need to find density
p = m/v What mass should I use? the weight of the scale for water, oil, or either?

You've calculated the density of the object and its volume. The only thing left to find is the mass.

 

[nick gear]

m= p * v => m = 9.3 g/cm^3 * 4.201 x 10 ^ -4 = .0039