A sphere is made up of two layers: The first is lead (density 11,340 kg/m³) and is of radius R. The second is tin, (7,310 kg/m³) and is concentric with the first, from radius R to radius 2R. The sphere is placed in a pool of mercury (density 13,600 kg/m³). How much of the volume of the sphere is below the surface? A) 68.5% B) 57.5% C) 57.0% D) 64.2%

Question

Quizplus · Accepted Answer

Let's assume the radius of the sphere is 1, so the volume of the sphere is:

V_sphere = (4/3)π(1)^3 = 4/3π

The volume of the lead layer is:

V_lead = (4/3)π(1)^3 - (4/3)π(1/2)^3 = 7/12π

The volume of the tin layer is:

V_tin = (4/3)π(2)^3 - (4/3)π(1)^3 = 7/3π

The total mass of the sphere can be calculated as:

m = ρ_lead*V_lead + ρ_tin*V_tin

where ρ_lead and ρ_tin are the densities of lead and tin respectively.

Substituting values, we get:

m = 11340*(7/12π) + 7310*(7/3π) = 27238.69 kg

The weight of the sphere can be calculated as:

w = m*g

where g is the acceleration due to gravity (9.8 m/s^2).

Substituting values, we get:

w = 27238.69*9.8 = 267101.962 N

The buoyant force on the sphere can be calculated as:

F_buoyant = ρ_mercury*V_sphere*g

where ρ_mercury is the density of mercury.

Substituting values, we get:

F_buoyant = 13600*(4/3π)*9.8 = 66794.74 N

The net force on the sphere is:

F_net = F_buoyant - w

Substituting values, we get:

F_net = 66794.74 - 267101.962 = -200307.22 N

Since the net force is downward, the sphere will sink until it displaces enough mercury to equal its weight. Let h be the height of the sphere below the surface of the mercury. Then the volume of the displaced mercury can be written as:

V_displaced = πh^2(3R - h)/3

The weight of the displaced mercury is:

w_displaced = V_displaced*ρ_mercury*g

Since the sphere is in equilibrium, the weight of the displaced mercury must be equal to the weight of the sphere:

w_displaced = w = 267101.962 N

Substituting values, we get:

V_displaced*13600*9.8 = 267101.962

Solving for h, we get:

h = 0.4257 m

Thus, the amount of the volume of the sphere below the surface is:

(4/3)π(1)^3 - π(0.4257)^2(3 - 0.4257)/3 = 0.575 or 57.5% approximately.

Therefore, the answer is B.

A Sphere Is Made Up of Two Layers: the First