What Is the Density of a Sphere Submerged in Water?

[05holtel]

Homework Statement

A sphere completely submerged in water is tethered to the bottom with a string.?
The tension in the string is one-third the weight of the sphere. What is the density of the sphere?

Homework Equations

The Attempt at a Solution

Fnet = 0 = Fb-T- mg

T=(Density of water x Volume of water x gravity) - (density of sphere x Volume of sphere x gravity)

T= (denisty of water - density of sphere) (Volume of object)(Gravity)

where volume of water = volume of sphere is completely submeged object

Not sure what to do now

Please help

 

[Doc Al]

The Attempt at a Solution

Fnet = 0 = Fb-T- mg

You are told what T is, so substitute what T is into that equation.

What's the buoyant force equal?

Hint: You want to rearrange the equation to solve for density, which is m/V.

 

[thomate1]

.

I would approach this problem by first understanding the concept of buoyancy and Archimedes' principle. This principle states that an object submerged in a fluid experiences an upward force equal to the weight of the fluid it displaces. In this case, the sphere is completely submerged in water, so the upward buoyant force is equal to the weight of the water it displaces.

We also know that the tension in the string is one-third the weight of the sphere. This means that the downward force of gravity on the sphere is three times greater than the upward buoyant force. We can express this mathematically as:

Fnet = 0 = Fb - T - mg

Where Fb is the buoyant force, T is the tension in the string, and mg is the weight of the sphere.

Since we are given that T = 1/3 mg, we can substitute this into the equation:

Fnet = 0 = Fb - (1/3)mg - mg

We can rearrange this to solve for the buoyant force:

Fb = (4/3)mg

Now, we can use the definition of buoyant force to find the volume of water displaced by the sphere:

Fb = (density of water)(volume of water)(gravity)

Substituting in the known values, we get:

(4/3)mg = (density of water)(volume of sphere)(gravity)

And since the volume of water displaced is equal to the volume of the sphere, we can write:

(4/3)mg = (density of water)(volume of sphere)(gravity)

Solving for the density of the sphere, we get:

Density of sphere = (1/3)(density of water)

So, the density of the sphere is one-third the density of water. This makes sense, as the sphere is floating in water, indicating that it is less dense than water. In conclusion, the density of the sphere is one-third the density of water.