Question

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

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

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 \(V = \frac{4}{3}\pi r^3\).

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.

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.

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.

Key concepts

Archimedes' Principle

Archimedes' Principle is a fundamental concept in physics that explains the buoyant force. This principle states that any object, when immersed in a fluid, experiences an upward force that is equal to the weight of the fluid displaced by the object. This is the reason why objects appear to be lighter in water. Understanding Archimedes' Principle helps in solving problems related to objects submerged in fluids, such as the basketball in this exercise.

Consider a basketball pushed underwater in a swimming pool. The buoyant force acting on it can be calculated by knowing the volume of the basketball and the density of the fluid (water, in this case). The formula for buoyant force is given as:

• \(F_b = \rho_{\text{fluid}} V g\)

where \(F_b\) is the buoyant force, \(\rho_{\text{fluid}}\) is the density of the fluid, \(V\) is the volume of the submerged object, and \(g\) is the acceleration due to gravity.
In this context, Archimedes' Principle helps us calculate how much upward force the basketball experiences, allowing us to determine how buoyant it is in water.

Volume of a Sphere

The volume of a sphere is a crucial concept when calculating the buoyant force, as it determines how much fluid the object displaces. The formula for the volume of a sphere is

• \(V = \frac{4}{3} \pi r^3\)

where \(V\) is the volume and \(r\) is the radius of the sphere.

The exercise gives the circumference of the basketball, so to find the radius, you can use the formula:

• \(C = 2\pi r\)

Rearranging gives \(r = \frac{C}{2\pi}\). Once you have \(r\), plug it into the volume formula to get the volume of the basketball.

This volume is essential for calculating any forces that depend on fluid displacement, such as the buoyant force using Archimedes' Principle.

Drag Force

Drag force is a type of resistance that an object encounters when it moves through a fluid. It acts in the opposite direction of the object's motion, slowing it down. In the scenario where a basketball is rising through water, the drag force will work against the upward motion caused by buoyancy.

When the basketball reaches a constant velocity as it rises, it implies a balance of forces. The buoyant force pushing it upward is equal to the sum of the drag force and the force of gravity pulling it downward.

• The equation representing this force balance is:\(F_b = F_g + F_d\)
• Solving for drag force gives: \(F_d = F_b - F_g\).

Here, \(F_d\) represents the drag force, \(F_b\) is the buoyant force, and \(F_g\) is the force of gravity on the basketball. By knowing these values, one can precisely calculate the drag force acting on the basketball.

Force of Gravity

The force of gravity on an object is commonly known as its weight. It can be calculated by multiplying the mass of an object by the gravitational acceleration, which is approximately \(9.81 \, m/s^2\) near the Earth's surface. The formula is:

• \(F_g = mg\)

where \(F_g\) is the force of gravity, \(m\) is the mass, and \(g\) is the gravitational acceleration.

In this exercise, the mass of the basketball is given in grams. It should be converted to kilograms (\(1 \, ext{kg} = 1000 \, ext{g}\)) for use in the formula. Once you have the mass in kilograms, multiplying it by \(9.81 \, m/s^2\) gives you the gravitational force acting on the basketball.

This force plays a vital role in counterbalancing the buoyant force and is needed to determine the net force acting on the basketball. Working with these forces together allows one to understand the overall dynamics of objects submerged and moving in fluids.