Franky claims he has a mass of 65kg. He is sitting on a spherical bouy that
�oats just barely submerged under water. When he gets o� it rises halfway out
of the water. What must the radius R in centimeters of the sphere be for him
to be telling the truth?
To solve this problem, we need to understand the concept of buoyancy and Archimedes' principle.
Archimedes' principle states that when an object is submerged in a fluid, it experiences an upward buoyant force equal to the weight of the fluid it displaces. This means that for the object to float, the buoyant force must be equal to the weight of the object.
In this case, the object is the spherical buoy and the fluid is the water. Since the buoy floats just barely submerged, we know that the buoyant force is equal to the weight of the buoy. Let's call the weight of the buoy W.
The weight of an object can be calculated using the formula: W = m * g, where m is the mass of the object and g is the acceleration due to gravity (approximately 9.8 m/s^2).
In this case, Franky claims to have a mass of 65 kg. Therefore, the weight of the buoy is W = 65 kg * 9.8 m/s^2 = 637 N.
Now, let's find the volume of water displaced by the buoy. Since the buoy rises halfway out of the water, it displaces a volume of water equal to half of its own volume.
The volume of a sphere can be calculated using the formula: V = (4/3) * π * R^3, where V is the volume and R is the radius of the sphere.
Therefore, the volume of water displaced by the buoy is V_water = (1/2) * (4/3) * π * R^3 = (2/3) * π * R^3.
Now, according to Archimedes' principle, the weight of the water displaced should be equal to the weight of the buoy. Let's call the density of water ρ_water.
The weight of the water displaced can be calculated using the formula: W_water = V_water * ρ_water * g, where ρ_water is the density of water (approximately 1000 kg/m^3).
Since we want to find the radius R in centimeters, we need to convert the density of water to kg/cm^3. There are 1000 kg in 1 cubic meter, and there are 10^6 cm^3 in 1 cubic meter. Therefore, 1 kg/m^3 = 10^-6 kg/cm^3.
Now, let's substitute the values into the equation: W_water = (2/3) * π * R^3 * (1000 kg/m^3 * 10^-6 kg/cm^3) * 9.8 m/s^2.
Simplifying the equation, we have: W_water = (2/3) * π * R^3 * 0.0098 kg/cm^3.
Since the weight of the water displaced should be equal to the weight of the buoy, we can set them equal to each other and solve for R.
637 N = (2/3) * π * R^3 * 0.0098 kg/cm^3.
Dividing both sides of the equation by [(2/3) * π * 0.0098 kg/cm^3], we get:
R^3 = (637 N) / [(2/3) * π * 0.0098 kg/cm^3].
Taking the cube root of both sides of the equation, we can solve for R to find the radius of the sphere in centimeters.
R = ([(637 N) / [(2/3) * π * 0.0098 kg/cm^3])^(1/3).
Evaluating the expression on the right side of the equation will give us the value of R in centimeters.