Problem 1 - University Physics with Modern Physics

Question

Salt water has a greater density than freshwater. A boat floats in both freshwater and salt water. The buoyant force on the boat in salt water is that in freshwater. a) equal to b) smaller than c) larger than

Step-by-Step Solution

Answer

b) smaller than the buoyant force on the boat in freshwater. c) larger than the buoyant force on the boat in freshwater.

Step 1: Understand the Archimedes' Principle for buoyant force

According to Archimedes' principle, the buoyant force acting on an object submerged in a fluid (in this case, water) is equal to the weight of the fluid displaced by the object. Mathematically, it is given by: Buoyant force (F_b) = (density of fluid) * (volume of fluid displaced) * (acceleration due to gravity).

Step 2: Compare the densities of saltwater and freshwater

Salt water has a greater density than freshwater due to the presence of dissolved salts. Let's denote the density of saltwater as ρ_s and the density of freshwater as ρ_f, so ρ_s > ρ_f.

Step 3: Analyze the boat's displacement in both fluids

To float, a boat must displace an amount of fluid that has a weight equal to the boat's weight. The volume of fluid displaced by the boat in both cases will be different since the densities are different. In saltwater, the boat has to displace less fluid volume because of the higher density to maintain equilibrium (compared to freshwater). Let's denote the volume of fluid displaced in saltwater as V_s and in freshwater as V_f, so V_s < V_f.

Step 4: Calculate the buoyant forces in saltwater and freshwater

Now, we can find the buoyant forces in both cases. For saltwater, the buoyant force is F_bs = ρ_s * V_s * g, and for freshwater, the buoyant force is F_bf = ρ_f * V_f * g.

Step 5: Compare the buoyant forces

Since the density of saltwater is greater than that of freshwater (ρ_s > ρ_f) and the volume displaced in saltwater is less than that in freshwater (V_s < V_f), it might be unclear how these factors affect the buoyant force directly. However, recall that the boat's weight remains constant in both cases. The buoyant force must be equal to the boat's weight to maintain equilibrium, so the buoyant force remains the same in both scenarios. The buoyant force on the boat in salt water is: a) equal to the buoyant force on the boat in freshwater.