In seawater, a life preserver with a volume of 0.0400 m^3 will support a 75.0 kg person (average

In seawater, a life preserver with a volume of 0.0400 m^3...

In seawater, a life preserver with a volume of 0.0400 $\mathrm{m}^{3}$ will support a 75.0 $\mathrm{kg}$ person (average density 980 $\mathrm{kg} / \mathrm{m}^{3} )$ with 20$\%$ of the person's volume above water when the life preserver is fully submerged. What is the density of the material composing the life preserver?

In seawater, a life preserver with a volume of 0.0400 $\mathrm{m}^{3}$
will support a 75.0 $\mathrm{kg}$ person (average density 980 $\mathrm{kg} / \mathrm{m}^{3} )$ with 20$\%$ of the person's volume above water when the life
preserver is fully submerged. What is the density of the material composing the life preserver?

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Key Concepts

Archimedes' Principle

This principle states that the buoyant force acting on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. It is fundamental in understanding how objects float or sink, as it provides the relationship between the volume of displaced fluid and the forces acting on the object.

Density

Density is the measure of mass per unit volume and is a critical property when determining buoyancy. Comparing the density of an object with that of the surrounding fluid determines whether the object will float or sink, as a lower density generally indicates that the object will displace enough fluid to support its weight.

Buoyant Equilibrium

For an object to float, it must reach a state of buoyant equilibrium where the upward buoyant force balances the downward gravitational force. This equilibrium condition can be used to determine relationships between submersion fraction, the object's volume, and its mass, and it is essential in calculating the necessary material density in flotation devices.

Transcript

00:01 Okay, so in this problem we have a person using a life preserver on seawater.

00:06 So here is the seawater.

00:10 Here we have the person with 80 % of his body below the water.

00:23 The person is using the life preserver.

00:26 So let's draw the life preserver as this thing.

00:30 Or in here.

00:32 Okay.

00:32 And the problem wants to know, let's see, what is the density of the material composing the life preserver? so first of all, let's remember, let's draw the forces involving this problem.

00:50 First of all, we have a buoyant force in here, and we have the weight of the life preserver plus the weight of the person.

01:03 Okay, so this is going to be a capital m, g, because it's the mass of the person and the mass of the seawater.

01:13 I actually draw this wrong because the problem says that the life preserver is completely below the water.

01:24 So that is a much more precise draw.

01:29 Nice.

01:30 Okay, so let's see.

01:33 The equilibrium of forces in this problem is going to be the buoyant force equals m .g.

01:45 But who is the buoyant force? well, let's remember that buoyant force is just the density of the sea water, the volume of the objects will emerge, times the gravity's acceleration...

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