A plastic sphere floats in water with 50.0 percent of its volume submerged. This same sphere flo

A plastic sphere floats in water with 50.0 percent of its...

Key Concepts

Archimedes’ Principle

Archimedes’ Principle states that an object immersed in a fluid experiences an upward buoyant force equal to the weight of the fluid it displaces. This concept is fundamental in determining whether an object will float or sink and by how much of its volume will be submerged when in equilibrium.

Buoyancy and Density Relationship

The buoyant force acting on an object in a fluid is directly related to the density of the fluid and the volume of the fluid displaced. Comparing the object's density with that of the fluid determines the fraction of the object that will be submerged, as the object will float when its weight is balanced by the weight of the displaced fluid.

Equilibrium Condition for Floating Bodies

For an object floating in a fluid, the equilibrium condition requires that the weight of the object is equal to the buoyant force. This condition is used to establish a relationship between the object's density, the fluid's density, and the fraction of the object’s volume that is submerged, enabling calculations of unknown densities.

Fraction of Volume Submerged

The fraction of an object's volume that is submerged in a fluid is equal to the ratio of the object's density to the density of the fluid, provided the object is in a state of floating equilibrium. This concept allows one to deduce the densities involved by comparing the observed submerged fraction in different fluids.

Transcript

00:09 The following problem is given where a plastic sphere is submerged 50 % in this body of water, and there's also the same plastic sphere submerged 40 % into this glycerin liquid.

00:27 The point of this problem is to solve for the density of the glycerin and solve for the density of the sphere.

00:35 To start this problem, we'll start with a system of equations where the buoyancy force equation, buoyancy is equal to the density of the water times the volume of the water times the gravitational force.

01:00 And for the glycerin, the buoyancy force will be equal to density of glycerin times the volume of glycerin times gravitational force g.

01:24 Now we can set these equations equal to one another.

01:29 We have row water times the volume of the water times gravity, equalling row glycerin times volume of glycerin times gravity.

01:51 You can see in this equation gravity is going to cancel out, and we can solve by isolating for roguicin is what they asked for in the problem.

02:03 So we have roe glycerin equaling density of the water times the volume of the water over the volume of glycer.

02:21 And we know some of these values, so we can start plugging in for the density of glycerin to find that answer first.

02:30 So we know that the density of water is 1 ,000 kilograms per meter cubed.

02:55 So you can plug that value in, and then since it is only submerged 50 % in the water, we can multiply that by 0 .5 times the volume of the sphere, which is the, as in the previous equation, is volume in the water, but it is the same sphere in both situations.

03:22 And then the volume of in glycerin is only submerged 40%.

03:28 So we're going to multiply by 0 .4 times that volume of the sphere.

03:33 And you can see the volume of sphere is going to cancel out in both equations...

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