A sphere of uniform density d1 has a radius R. It also consists of a hollow sphere of radius r. the sphere floats slightly on water of density d2. Find r.
To find the value of r, we can use the principle of buoyancy.
Buoyancy is the upwards force exerted on an object submerged (or partially submerged) in a fluid, due to the difference in pressure between the top and bottom surfaces of the object. In the case of floating objects, the buoyant force is equal to the weight of the fluid displaced by the object.
First, let's calculate the weight of the sphere. The weight of an object can be calculated using the formula: weight = mass * g, where g is the acceleration due to gravity (approximately 9.8 m/s²).
The mass of a sphere can be calculated using the formula: mass = density * volume.
The volume of a solid sphere can be calculated using the formula: volume = (4/3) * π * r³.
Since the sphere consists of a hollow portion, we need to subtract the volume of the hollow portion from the total volume to get the volume of the solid portion.
The volume of the hollow portion can be calculated using the formula: volume = (4/3) * π * (r² - R²).
Once we have the weight of the sphere, we can calculate the weight of fluid displaced by the sphere using the formula: weight_displaced = density_water * volume_displaced.
Since the sphere is floating, the weight of the fluid displaced is equal to the weight of the sphere (buoyant force = weight of the fluid displaced).
Therefore, we can set up the equation: weight_displaced = weight_sphere.
Let's substitute the values and solve the equation to find the value of r.
Final equation: density_water * volume_displaced = density_sphere * (volume_solid - volume_hollow)
Substituting the formulas for volume and simplifying, we get:
density_water * (4/3 * π * r³) = density_sphere * (4/3 * π * r³ - 4/3 * π * (r² - R²))
Canceling out common terms and simplifying further, we get:
density_water * r³ = density_sphere * r³ - density_sphere * (r² - R²)
Rearranging the equation, we obtain:
density_sphere * (r² - R²) = density_water * r³
Finally, solving for r, we get:
r = [density_sphere * R²] / [density_sphere - density_water]
Substituting the values of density_sphere = d1 and density_water = d2, we can calculate the value of r using their respective densities and the given values of R and d1.
I hope this explanation and the provided formula help you find the value of r in the given scenario.