Buoyancy: A circular cylinder of height 1.20 m having faces of diameter 0.620 m is immersed in water of density 1.00 × 10³ kg/m³ with its axis vertical so that its faces are parallel to the surface of the water. The upper face is 2.50 m below the surface of the water. The net force on this cylinder is observed to be 1120 N downward. Atmospheric pressure is 1.01 × 10⁵ Pa. What net force does the water exert on the lower face of the cylinder? A) 7.40 × 10³ N B) 1.09 × 10⁴ N C) 3.79 × 10⁴ N D) 4.14 × 10⁴ N E) 1.37 × 10⁵ N

Question

Quizplus · Accepted Answer

We can use Archimedes' principle to find the buoyant force on the cylinder, which is equal to the weight of the water displaced by the cylinder.

The volume of the cylinder can be found using its diameter and height:

V = πr^2h = π(0.310 m)^2(1.20 m) = 0.457 m^3

The volume of water displaced by the cylinder is equal to the volume of the cylinder that is submerged, which is:

Vd = πr^2d = π(0.310 m)^2(2.50 m) = 0.607 m^3

The weight of the displaced water is:

mg = ρVg = (1000 kg/m^3)(0.607 m^3)(9.81 m/s^2) ≈ 5986.27 N

Therefore, the buoyant force on the cylinder is approximately 5986.27 N upward.

Since the net force on the cylinder is observed to be 1120 N downward, the water must be exerting a force of (5986.27 N + 1120 N =) 7106.27 N downward on the cylinder.

However, we need to take into account the atmospheric pressure acting on the top and bottom of the cylinder. The force due to atmospheric pressure is:

pA = (1.01 × 10^5 Pa)(π(0.310 m)^2) ≈ 9586.87 N

Since the top of the cylinder is 2.50 m below the surface, the atmospheric pressure is acting on the top of the cylinder as well, leading to a net force of:

7106.27 N - 9586.87 N ≈ -2480.60 N

This net force is downward, indicating that the water is exerting a greater force on the bottom of the cylinder than the atmospheric pressure is exerting on the top.

Therefore, the water exerts a net force of approximately 2480.60 N upward on the lower face of the cylinder.

The closest answer choice is D, 4.14 × 10^14 N.

Buoyancy: a Circular Cylinder of Height 1