Rubber ducky, you’re the one.

A 10-g rubber ducky floats in a tub of water at bath time.

Draw and label a force diagram for the ducky.

How much of the volume of the ducky must be submerged in order for it to float?

Fbuoyant = ρfluidgVsubmerged

1) The buoyant force is directly proportional to the density of the fluid.

- More dense fluids exert a greater buoyant force!

2) The buoyant force is also directly proportional to the amount of volume that is submerged.

- The buoyant force depends on the volume of liquid that is

displaced by the object.

Volume submerged = volume of fluid displaced

The volume of the crown will be equal to the volume of the displaced water when the crown is submerged!

Volume of crown

Fbuoyant = ρfluidgVsubmerged

Fbuoyant = ρfluidgVdisplaced fluid

But, we can use the definition of density to simplify!

Fbuoyant = mdisplaced fluid*g

The buoyant force is equal in magnitude to the weight of the displaced fluid!

The buoyant force is equal to the weight of the fluid that is displaced.

The scale has to pull 2 N less, since

the water is exerting an upward buoyant

force of 2 N.

Flotation – How much of an object will be submerged?

If an object is floating, the buoyant force is exactly balanced with the gravitational force.

Ffluid on object

Fearth on object

Gravitational forcemobjectg

Gravitational forceρobjectVobjectg

Buoyant forceρfluidVsubmergedg

Buoyant forceρfluidVsubmergedg

But, we can use the definition of density for a more

interesting comparison!

If the object is floating…

Gravitational force = Buoyant force

ρobjectVobjectg = ρfluidVsubmergedg

ρobjectVobject = ρfluidVsubmerged

When an object is floating on top of a fluid.

Flotation: Fun with Ratios!

The ratio of the density of the object to the density of the fluid will be the fraction of the object that will be submerged when it floats!

Caution: This only applies if the object is floating! It does not apply if an object is completely submerged.

Ice is 9/10 as dense as water, therefore 9/10 of an iceberg will be submerged!

This same ratio will also apply to ice cubes floating in water.

1) A cube of side length L is made of a substance that is ¼ as dense as water. When placed in a calm water bath, the cube will 

(A) float with ½ L above the surface.(B) sink to the bottom.(C) float with ¼ L above the surface.(D) float with ¼ L below the surface.(E) None of the above

2) An ice cube floats in a glass of lemonade so that ¾ of it is submerged. If L is the density of the lemonade, then the density of the ice cube is  (A) L/4 (B) L/3 (C) 3L/4 (D) L (E) 4L/3

Flotation Whiteboards – 2 for 1 special!

If the object is less dense than the fluid, then the fractions will be less than one.

This means that not all of the object is submerged when it is in equilibrium.

AKA - It floats!!

If the object is more dense than the fluid, the fractions will be more than one.

This is nonsense, and means that the equation does not apply.

AKA – It sinks!!

This only applies if the object is less dense than the fluid! It does not apply if an object will sink.

A hot air balloon is filled with helium to a volume of 400 m3, and attached to a spring scale so that it doesn’t float away. The balloon and its helium contents have a combined mass of 300 kg. The density of air at sea level is 1.23 kg/m3.

Buoyant force keeps balloons afloat!

Draw a force diagram for the balloon.

Determine the scale reading.

Fair on balloon

(buoyant)

Fearth on balloon

Fscale on balloon

Fair on balloon (buoyant) = Fscale on balloon + Fearth on balloon

Remember – A scale tells you how much force IT exerts!

Bonus Extra Credit Question!

Is this plausible? Look up some of the numbers, make some reasonable estimates, and come up with a supported argument for or against its feasibility.

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