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Chapter 13: Problem 56

A basketball of circumference \(75.5 \mathrm{~cm}\) and mass \(598 \mathrm{~g}\) is forced to the bottom of a swimming pool and then released. After initially accelerating upward, it rises at a constant velocity, a) Calculate the buoyant force on the basketball. b) Calculate the drag force the basketball experiences while it is moving upward at constant velocity.

Short Answer

Expert verified
Question: Calculate the buoyant force and drag force acting on a basketball submerged in a swimming pool. The basketball has a circumference of 0.75 meters and a mass of 600 grams. The density of water is 1000 kg/m³. Answer: To calculate the buoyant and drag forces on the basketball, follow the steps outlined in the solution. In Step 1, find the volume of the basketball using its circumference. In Step 2, calculate the buoyant force using the Archimedes' principle. In Step 3, calculate the force of gravity on the basketball. Finally, in Step 4, calculate the drag force by balancing the buoyant force and the force of gravity when the basketball is moving at a constant velocity.

Step by step solution

01

Find the volume of the basketball

Using the given circumference of the basketball, we can find its radius and then calculate its volume. The formula for the circumference of a sphere is \(C = 2\pi r\). Therefore, the radius can be found by rearranging this equation as \(r = \frac{C}{2\pi}\). Hence, the volume of the basketball can be calculated using the formula for the volume of a sphere, which is \(V = \frac{4}{3}\pi r^3\).
02

Calculate the buoyant force

The buoyant force can be calculated using the Archimedes' principle, which states that the upward buoyant force exerted on a body immersed in a fluid is equal to the weight of the fluid that the body displaces. The buoyant force formula is \(F_b = \rho_{fluid}Vg\), where \(\rho_{fluid}\) is the density of the fluid, \(V\) is the volume of the submerged object, and \(g\) is the acceleration due to gravity. Using the density of water as \(1000\, kg/m^3\), the volume of the basketball obtained in step 1, and the standard value for the acceleration due to gravity (\(9.81\, m/s^2\)), we can calculate the buoyant force on the basketball.
03

Calculate the force of gravity on the basketball

To calculate the force of gravity on the basketball, we can use the formula \(F_g = mg\), where \(m\) is the mass of the basketball and \(g\) is the acceleration due to gravity. We are given the mass of the basketball in grams, so we should convert it to kilograms before using it in the formula.
04

Calculate the drag force

When the basketball is moving upward at a constant velocity, the net force acting on it is zero. This means that the buoyant force and the force of gravity must be balanced by the drag force. We can calculate the drag force using the following equation: \(F_d = F_b - F_g\). With all of the information we have gathered in the previous steps, we can now perform the calculations needed to find the buoyant and drag forces on the basketball.

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Most popular questions from this chapter

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