spherical aluminum ball of mass 1.22 kg contains an empty spherical cavity that is concentric with the ball. The ball just barely floats in water.Calculate the radius of the cavity.

R = outer radius

r = inner radius (of spherical cavity)

Sphere weight = buoyancy force

(4/3) pi (R^3 - r^3)*(density of Al)*g
= (4/3) pi R^3 * (density of water)*g

(4/3) pi R^3 * (density of water)
= 1.22 kg

(Use the second equation to solve for R)

The (4/3) pi g cancels out in the first equation.

1 - (r/R)^3
= (water density)/(Al density)

Look up the densities and solve

To calculate the radius of the cavity, we can use Archimedes' principle and the concept of buoyancy.

Step 1: Determine the buoyant force acting on the aluminum ball.
The buoyant force on an object submerged in a fluid is equal to the weight of the fluid displaced by the object. In this case, the buoyant force is equal to the weight of the water displaced by the aluminum ball. Since the aluminum ball is floating, the buoyant force is equal to the weight of the ball itself.

Weight of the ball = mass * acceleration due to gravity
= 1.22 kg * 9.8 m/s^2

Step 2: Calculate the volume of the aluminum ball.
We can use the density of aluminum (2,700 kg/m^3) to determine its volume.
Density = Mass / Volume

Mass of the aluminum ball = 1.22 kg
Density of aluminum = 2,700 kg/m^3

Volume of the aluminum ball = Mass of the ball / Density of aluminum

Step 3: Determine the volume of the cavity.
Since the cavity is completely hollow and concentric with the ball, the volume of the cavity is equal to the volume of the aluminum ball.

Volume of the cavity = Volume of the aluminum ball

Step 4: Calculate the radius of the cavity.
We can use the formula for the volume of a sphere to find the radius.

Volume of a sphere = (4/3) * π * r^3

Since the volume of the cavity is equal to the volume of the aluminum ball, we can set their equations equal to each other.

(4/3) * π * r_cavity^3 = Volume of the aluminum ball

Now, we can solve for the radius of the cavity (r_cavity).

r_cavity = ∛((Volume of the aluminum ball) * (3/4π))

Substituting the previously calculated values, we can find the radius of the cavity.