04_39 Buoyant forces and densities

[AznBoi]

A spring scale calibrated in kilograms is used to determine the density of a rock specimen. The reading on the spring scale is 0.45 kg when the specimen is suspended in air and 0.36 kg when the specimen is fully submerged in water. If the density of water is 1000 kg/m^3, the density of the rock specimen is

A) 2.0 x 10^2
B) 8.0 x 10^2
C) 1.25 x 10 ^3
D) 4.0 x 10^3
E) 5.0 x 10^3

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Solution of book:
The buoyant force is equal to the weight of the water displaced and is the difference between the apparent weight and actual weight. So the mass of the water displaced is the diff between the actual and apparent masses... or 0.45 - 0.36 = 0.09 kg is mass of water displaced.

the water displaced and the rock have the same volume, so the ratio of their densities is the same as the ratio of their masses. Ratio of masses is

0.45 / 0.09 = 5.0 the rock is five times the mass, so must be five times denser than water.

Density of water is 1000 kg/m3 so density of rock is 5000 kg/m3


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I've used a method to get the answer, and since the specimen is complete submerged in the water the volumes are the same. (am I right so far?)

So then you can combine the volumes and solve for the actual density of the specimen: = 5000kg/m^3
Is this a logical/correct way of solving this problem?

I know that using ratios and such is much more efficient in a lot of situations where the problem gives no values for unknowns except for ratios. How do you know what ratio to use and how do you get better at using them? I'm confused about the solution's method and would like to learn it, thanks!

 

[Mindscrape]

I would use Newton's second law. You know that the buoyancy force is F_b = \rho V g. You also know, by Newton's second law and conservation laws, that the force of gravity minus the buoyant force will be the displaced force F_g - F_b = F_a. At which point the density of the object will be the ratio of force of gravity to the buoyancy force, which will be the difference of the force of gravity and the apparent force of gravity (apparent weight). So \rho_{rock} = \frac{F_g}{F_b} = \frac{F_g}{F_g-F_a}. Your textbook just factored out the gravity term since it appears in every term.