LECTURE 3 BUOYANCY (ARCHIMEDES’ PRINCIPLE) Lecture Instructor: Kazumi Tolich Lecture 3

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¨ Reading chapter 15.4 and 15.5 ¤ Archimedes’ principle and buoyancy Buoyancy and Archimedes’ principle

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¨ The force exerted by a fluid on a body wholly or partially submerged in it is called the buoyant force.

¨ Archimedes’ principle:

An object completely immersed in a fluid experiences an upward buoyant force equal in magnitude to the weight of fluid displaced by the object.

� = ���

� is the volume of displaced fluid by the object. Demo: 1

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¨ Archimedes’ principle ¤ The buoyant force is equal to the weight of the water displaced. Quiz: 1

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¨ Suppose that a rock is sitting on a chair, both submerged in water. Another identical rock is sitting on a chair in the air. Which chair exerts greater normal force on the rock? A. The chair in the water B. The chair in the air C. Actually, both chairs exert the same amount of force. Quiz: 3-1 answer

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A. The chair in the air

¨ The rock is sitting, so it is not accelerating. The net force on it is zero.

¨ The rock has the following forces: buoyant force by the fluid (upward), normal force by the chair (upward), and weight (downward) � ¨ The weight of the rocks are the same since they are identical. � ¨ The buoyant force by the air is smaller because air is less dense while � the volume of the rocks are identical: �, = ��� < �, = ���.

¨ This is why it is easier to lift a rock (or any object) in water than in air. The crown and the nugget 7

Archimedes (287-212 BC) had been given the task of determining whether a crown made for King Hieron II was pure gold. In the above diagram, crown and nugget balance in air, but not in water because the crown has a lower density. Quiz: 2

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¨ An inverted glass jar, filled with air, sits on a table. A helium balloon inside the jar floats to the top. If the air is replaced with helium, what will happen to the helium balloon? A. it still floats at the top because it has a upward buoyant force. B. it stays in the middle because it has zero net force. C. it sinks to the bottom because of the weight of the balloon. D. the balloon shrinks in size due to the surrounding helium. E. the balloon grows in size due to the lack of surrounding air. Quiz: 3-2 answer

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¨ it sinks to the bottom because of the weight of the balloon.

¨ The balloon floats initially because the displaced air weighs more than the balloon, so the buoyant force provides a net upward force.

¨ When the balloon is in the lighter helium gas (instead of air), the displaced helium gas provides a buoyant force equal to the weight of the gas in the balloon.

¨ But it does not provide enough upward buoyant force to support the weight of the balloon. Quiz: 3

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¨ Now the jar is lifted off the table, but the jar remains inverted to keep the helium gas in the jar. What will happen to the balloon? A. It floats at the top of the jar. B. It floats so that the bottom of the balloon is near the bottom of the jar. C. It floats so that the top of the balloon is near the bottom of the jar. D. It sinks down to the surface of the table. Quiz: 3-3 answer

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¨ It floats so that the bottom of the balloon is near the bottom of the jar.

¨ The balloon sinks in the helium gas until it hits the surface of the air.

¨ Because the balloon floats in air, it will float on the surface of the air and therefore remain mostly inside the jar, but at the bottom. Demo: 2

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¨ Helium balloon in helium

¨ Helium balloon in liquid nitrogen ¤ Demonstration of buoyancy and Archimedes’ principle Example: 1

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¨ A block of an unknown material’s apparent weight is � = 5.00 N in air and � = 4.55 N when submerged in water. What is the density of the material? Quiz: 4

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¨ Which of the following statements is/are correct? Choose all that apply.

A. For an object to float in water, its density must be less than that of water.

B. For an object to float in water, its density must be more than that of water.

C. For an object to float, it must displace water whose volume is the same as the volume of the object.

D. For an object to float, it must displace water whose weight is the same as the weight of the object. Quiz: 3-4 answer

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¨ For an object to float, it must displace water whose weight is the same as the weight of the object.

¨ For an object to float, the net force on it must be zero since it is not accelerating vertically.

¨ The the magnitude of the buoyant force must be equal to the magnitude of the weight.

¨ � = ���, where � is the volume of displaced fluid by the object.

¨ An object that is denser than fluid can float if it displaces enough fluid because of its shape. Demo: 3

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¨ Battleship

¨ Board and weights Quiz: 5

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¨ Two beakers, a and b, are filled to the brim with water. A wooden block is placed in beaker b so that it floats. (Some of the water will overflow the beaker). Both beakers are then weighed. Which scale reads a larger weight? Quiz: 3-5 answer

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¨ Same for both

¨ Since the wooden block floats, the net force on the block is zero.

¨ The wooden block displaces an amount of water whose weight is the same as the weight of the block. Example: 2

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¨ What is the minimum area of the top surface of a slab of ice with thickness, ℎ = 0.30 m, floating on fresh water that will hold up an automobile of mass � = 1100 kg sitting on top? The density of water and ice 3 are � = 0.998 × 10 kg/m 3 and � = 0.917 × 10 kg/m , respectively.