AP Physics B – Simple and Interesting Fluid Mechanics

“There is no democracy in physics. We can't say that some second-rate guy has as much right to an opinion as Fermi.”

–Luis Walter Alvarez

The first free response question in the AP Physics B 2013 Examination was from fluid mechanics which provides ample scope for question setters to ask interesting questions. Today we shall discuss some interesting and useful points in this section.

An important point you need to note is that an object denser than a liquid, can displace liquid of  weight equal to that of the object, if the object forms part of a floating  body.  The same object submerged independently in the liquid will displace liquid of weight less than that of the object.

(1) To make the above point more clear, let us consider a large tank containing water of density 1000 kg/m³. A wooden block A of specific gravity 0.5 and volume 2 m³ is floating in the water (Fig. 1). An object B of specific gravity 5 and volume 0.1 m³ is placed on the wooden block. Will the wooden block still continue to float?

The wooden block A has mass 1000 kg as is clear from the relation,

             mass = volume×density

[Since the specific gravity of the wooden block is 0.5, its density is 0.5×1000 = 500 kg/m³. Therefore, the mass of the wooden block = 2×500 = 1000 kg.

Remember that the specific gravity or relative density of a body is the ratio of the density of the body to the density of water. Since water has density 1000 kg/m³ in the international system (SI) of units, the density of a body has numerical value 1000 times its specific gravity].

Mass of the object B placed on the wooden block is 0.1×5000 = 500 kg. Therefore the total mass of the wooden block and the object placed on it is (1000+500) kg = 1500 kg.

Since the volume of the wooden block is 2 m³ it can displace 2 m³ of water before getting submerged. The wooden block and the object placed on it will continue to float since the total volume of water to be displaced needs to be 1.5 m³ only.

[Mass of 1.5 m³ of water is 1500 kg].                                                                                    

In the above explanation we have used the law of floatation, which states that the weight of a floating body is equal to the weight of the displaced liquid.

(2) Now, consider a slightly different situation which is shown in the adjoining figure (Fig. 2). The same wooden block A and the denser object B as we considered above are shown here also; but the object is suspended from the wooden block using a rope of negligible mass. How much water will be displaced by the system (containing the wooden block and the object) in this case?

The object has specific gravity 5 and it is fully immersed in water. If it were not part of the floating system, it would have displaced water of volume equal to its own volume (which is 0.1 m³). Since the object is part of a floating system, it is capable of displacing water of mass equal to its own mass (which is 500 kg), by pulling the wooden block further down making use of the connecting rope. Thus the total mass of water displaced is 1500 kg and the total volume of water displaced is 1.5 m³, as in the case of the system shown in Fig. 1.

In Fig. 1 the water level in the tank is shown as L₁ where as in Fig. 2 the water level is shown as L₂. Obviously these two levels coincide.

(3) Now, suppose the rope connecting the object B to the wooden block A cut. The object B, being denser than water, will then sink and will rest at the bottom of the tank. How much water will be displaced by the wooden block and the object together in this case?

The wooden block is a floating body and hence it will displace water having mass equal to its own mass, which is 1000 kg. So the volume of water displaced by the wooden block is 1 m³. The object B will displace water of volume equal to its own volume, which is equal to 0.1 m³. Therefore, the total volume of water displaced by the wooden block and the object together in this case is 1.1 m³.

The water level in the tank will be lowered (compared to L₁) in this case.

(4) Let us once again consider the original system (Fig. 1) in which the denser object B (of specific gravity 5 and volume 0.1 m³) is placed on the wooden block. What must be the mass of the object B if the entire wooden block just begins to submerge?

Questions of the above type are often found in question papers of various examinations. When the wooden block is completely immersed in water the volume of water displaced is equal to the volume of the wooden block, which is 2 m³. The weight of the wooden block and the object together in this case is therefore equal to the weight of 2 m³ of water, which is 2000 kg. Since the mass of the wooden block is 1000 kg, the mass of the object must be 1000 kg.

We shall now consider another situation:

(5) Suppose the object placed on the wooden block (Fig. 1) has specific gravity 0.8 and the water level in the tank is noted. If the object is gently transferred to the water in the tank, will the water level in the tank change?

Since the specific gravity of the object is 0.8, it will float in water. Whether it is on the wooden block or independently outside, it will displace water of mass equal to its own mass. Therefore, there is no change in the water level in the tank.

(6) Now, suppose the object of specific gravity 0.8 (which we considered above) is tied to the wooden block using a rope of negligible mass so that it is fully immersed in the water in the tank, as shown in Fig. 3. Will the level of water in the tank change in this case?

Many among you might be confused about this situation. Understand that the wooden block and the object tied to it is still a floating system and the mass of water displaced must be equal to the mass of the floating system. Therefore there is no change in the water level in the tank.

We shall now consider the case of an ice block floating in water: 

(7) Consider a block of ice floating in a tank containing pure water (Fig. 4). If the ice melts without any change in the temperature, will the water level in the tank rise up, fall down or remain unchanged?

This is a very popular question and most of you know the answer: The water level will remain unchanged. But how do you justify your answer?

The block of ice is a floating body and the volume of water displaced by it has weight equal to its own weight. When the ice melts, the eaxtra water produced should fill exactly the volume originally occupied by the immersed portion of ice. Therefore, the water level in the tank is unchanged.

(8) Suppose a metallic bob is placed on the ice block so that it still floats in the water in the tank (Fig. 5). If the ice melts without any change in the temperature, will the water level in the tank change?

When the ice alone is there, the water level in the tank remains unchanged when the ice melts, as we have seen above. When the metallic bob also is present, it sinks when the ice melts and it displaces water of volume equal to its own volume. This volume is certainly less than the volume of water it displaces when it is part of the floating system formed along with the ice block. The water level in the tank is therefore lowered when the ice melts (without any change in the temperature).

(9) Now, suppose the metallic bob in the above question is replaced by a wooden bob (which is lighter than water). If the ice melts without any change in the temperature, will the water level in the tank change?

In this case there will be no change in the water level since the wooden bob floats in the water in the tank when the ice melts, displacing the same volume of water as before.