You are out fishing, when you go to reach for the cooler, when you clumsily knoc

Question

You are out fishing, when you go to reach for the cooler, when you clumsily knock it into the water. The cooler is in the shape of a cube with a side-length 0.2 meters. The cooler plus sandwiches inside weigh 6 pounds (26.7 N).

a) How deep below the water's surface is the bottom of the cooler, assuming it floats stationary with the top and bottom level with the water's surface?

b) When the cooler was just sitting on the boat, there was actually a buoyant force on it from air. How big is the buoyant force of air on the cooler? How many times smaller is that force than the weight of the cooler? Assume you are at sea level where the density of air is 1.225 kg/m^3

Explanation / Answer

a) By Archimedes principle, we know that for an object in water, the buoyant force is equal to the weight of the fluid displaced by the object.

In this case, the cooler is floating, so the net force is zero. The floating object displaces fluid whose weight would be the weight of the object, since the object fills the volume that was previously occupied by the fluid.

1. Buoyant force of the fluid = weight of the object

Density(w)*V(submerged)*g = Density(obj)*V(obj)*g

Rearranging the above equation,

V(sub)/V(obj) = Density(obj)/Density(w) ––>(1)

2. We can write the volume as,Volume = Area*height

Rewriting equation (1),

Area*h(sub)/Area*h(obj) = Density(obj)/Density(w)  

Or

h(sub)/h(obj) = Density(obj)/Density(w) ––> (2)

3. We know that Density of water is approx 1000kg/m^3. We have to find the density of the object.

Weight of object =mg= Density(obj)*V(obj)*g

i.e. 26.7 N = Density of the cooler*(0.2)^3 * 9.8

Density of the cooler = 26.7/(9.8*0.008) = 340.56 kg/m^3

4. To find the height of the cooler below the water surface

Substituting our unknowns in equation (2),

h(sub) = density(obj) / density(f)*h(obj)

= 340.56/1000*0.2

= 1.703m

So the height of the cooler that is submerged is 1.703m.

b) Bouyant force of air = Density(air)*V(obj)*g

= 1.225*(0.2)^3*9.8

= 0.096 N

Weight of the cooler is 26.7N

Taking the ratio,

buoyant force of air/weight of cooler = 0.096/26.7 = 3.6e-3

Or

buoyant force of air = 3.6e-3 * weight of cooler

Buoyant force of air is approx 0.004 times smaller than the weight of the cooler

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