Q&A: Finding Volume of a Floating Object

The following question was asked in the comments section of the article Archimedes' Principle Explained.

Q: Can you just explain how to measure the volume of an irregularly shaped object which floats in water? Can we use Archimedes' principle? How?


A: It is certainly possible to do so using Archimedes' principle, provided the density of the object is known. The key here is to find the mass of the object. Then we use the mass formula Mass = Vρ (where V is the volume of the object and ρ is the density of the object) to find the volume.

In my article Archimedes' Principle Explained, there is a section about objects floating freely. It is mentioned that

Weight of floating body = Weight of fluid displaced

therefore

Mass of floating body = Mass of fluid displaced

So if we can know the mass of water displaced (e.g. measuring the apparent increase in volume of water and multiplying it with its density, as in finding mass using the mass formula) when the object floats in water, we can know the mass of the object.

By rearranging the mass formula and substituting the mass and ρ into it, we can have the volume of the object. However, we must know the value of ρ beforehand.

Note: The above answer does not take into account other ways of finding the volume of an object that floats in water other than using Archimedes' principle.

Problem: Materials #4

A cube of ice is floating in a salt solution of density 1.25 times the density of pure water in a beaker. When the ice melts, the level of the solution in the vessel
A. rises
B. falls
C. remains unchanged
D. falls at first and then rises to the same height as before

Solution

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The answer is A.

A non-mathematical explanation would be that since the ice is floating, its mass would be the same as the mass of salt water displaced by it, as per Archimedes’ principle. Now consider the ice melted where the resulting water gradually “fills in” the space in the solution once occupied by the ice. The mass of the resulting water would still be the same as the mass of salt water displaced earlier, but the resulting water will occupy a larger space compared to the salt water displaced, because its density is lower (density is mass per unit volume). Hence the solution level rises.

A mathematical proof is as follows:

Mass of displaced salt solution = Mass of resulting water


This shows that the resulting water occupies more space compared to the displaced salt water in the ratio 5:4.

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Question source: Malaysia National Physics Competition 2007 (Secondary Level)

Problem: Materials #3

Within a certain type of star called a neutron star, the material at the centre has a mass density of 1.0 × 10¹⁸ kg m^-3. If a small sphere of this material of radius 1.0 × 10^-5 m were somehow transported to the surface of the earth, what would be the weight of this sphere? (Assume that g = 10 m s^-2)
A. 1000 N
B. 4200 N
C. 4.2 × 10⁴ N
D. 7.0 × 10⁴ N
E. 3.1 × 10⁹ N

Solution

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The answer is C.

From the information given, we need to first find the volume of this sphere, then find its mass and lastly calculate its weight. Therefore, we need the following formulas:


Weight, W = mg

where r is the radius of the sphere.


Step 1- Finding the volume of the sphere:


Step 2 - Finding the mass of the sphere:
The second formula can be rearranged to m = Vρ, so

m	= (4.189 × 10-15 m3)(1.0 × 1018 kg m-3) = 4.189 × 103 kg

Step 3 - Finding the weight of the sphere:

W	= (4.189 × 103 kg)(10 m s-2) = 4.189 × 104 kg m s-2 = 4.189 × 104 N

Rounding off, we have W = 4.2 × 10⁴ N.

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Question source: Malaysia National Physics Competition 2007 (Secondary Level)

Problem: Kinematics #4

A train covers half the distance of its journey with a speed 20 m s^-1 and the other half with a speed of 40 m s^-1. The average speed of the train during the whole journey is
A. 25 m s^-1
B. 27 m s^-1
C. 30 m s^-1
D. 32 m s^-1
E. 35 m s^-1

Solution

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The answer is B.

Let the total distance travelled by the train be d and the time taken for it to complete the first half and second half of the journey be t₁ and t₂ respectively. The distance travelled each half-journey is d/2 and it is the product of their respective travelling velocity and time taken.

For the first half-journey,


For the second half-journey,


Combining the above two equations, we get



Now we have the relationship between t₁ and t₂, we can eliminate one of them from our calculations. For convenience, let us state the d in terms of t₂ and find the average speed, which is the total distance travelled divided by the total time taken.



Rounding off to the nearest integer, we have 27 m s^-1.

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Question source: Malaysia National Physics Competition 2007 (Secondary Level)

The Concept of… Force

Force is one of the most fundamental concepts of physics. The concept of force has formed part of statics and dynamics since ancient times. The modern concept of force is commonly explained in terms of Newton's three laws of motion set forth in his Principia Mathematica (1687).


What is force?

Force can be described as a push or pull. A force upon an object is always a result of interaction with another object. When we say an object is acted by a force, there must be another object applying a force to it.

Force is also a vector, that is, it has magnitude and direction. We can fully describe a force by saying how strong it is (its magnitude) and in which direction it is acting. The SI Unit for force is newton (named after none other than Isaac Newton) and its symbol is N.


Types of Force

Forces can be categorised into two broad categories:

• Contact forces are those types of forces which result when two interacting objects are perceived to be physically interacting with each other. Examples of contact forces include frictional forces, tensional forces and applied forces.
• Action-at-a-distance forces are those types of forces which result even when the two interacting objects are not in physical contact with each other, yet are able to exert a push or pull despite their physical separation. Gravitational forces, magnetic forces and electrical forces are examples of this type of force.

The Effects of Force

Basically, a net force that acts on an object will produce an acceleration. For example, by applying a force, a stationary object can be set in motion, whereas a moving object can be made to move faster or slower and/or change its direction of motion (since acceleration also involves direction). In short, a force can alter the state of motion of an object; this is described in Newton’s first law of motion. The relationship between force and acceleration is further described in Newton’s second law of motion.

More often than not, a force that acts on an object does not cause the whole object to accelerate, but rather a part of it. Therefore a force can also cause the rotation and deformation of an object.

Photo by max's picks

When a force causes acceleration, it can transfer energy in the process. To illustrate this point, imagine a moving billard ball colliding with an 8-ball, as shown in the photo above. The billard ball will exert a force on the 8-ball so that the 8-ball accelerates and its velocity increases gradually. This increase in velocity in turn results in an increase in its kinetic energy. For this interaction to obey the law of conservation of energy, the object supplying the force must lose energy, therefore the application of force in this case causes energy to be transferred.

However, the presence of force does not necessary result in the transfer of energy. Now imagine a ball hitting a wall and bouncing back (assume the wall does not absorb any energy from the ball). Although acted by a force, the wall does not move and the ball continues to move with the same speed in the opposite direction. The energy of the ball remains the same.


What Causes Force?

It is mentioned above that an object acting by a force undergoes acceleration. Conversely, an object decelerating exerts a force on another object. This apparent symmetry can be explained using Newton's third law of motion and is the cause of contact forces.

For action-at-a-distance forces to act on an object, the object merely has to be within the influence of a field, like gravitational field, magnetic field and electrical field. Depending on the properties of the object, it may react towards more than one field. The exact reason why a field exerts a force is unknown, but the fact is that this is a phenomenon of our Universe.

An object that exerts a force on another object has the potential of transferring work to it, as mentioned above. Therefore, it is safe to say that for an object to exert a force on another object, it must first have energy (note that mass is also energy – concentrated energy – in agreement with the concept of mass-energy equivalence).

References

• "force." Encyclopædia Britannica. 2007. Encyclopædia Britannica Online. 11 Aug. 2007 <http://www.britannica.com/eb/article-9034834/force>
• "Force," Microsoft® Encarta® Online Encyclopedia 2007. 11 Aug. 2007. <http://encarta.msn.com/encyclopedia_761555718/Force.html>