Question

A plastic beach ball has radius \(20.0 \mathrm{cm}\) and mass \(0.10 \mathrm{kg},\) not including the air inside. (a) What is the weight of the beach ball including the air inside? Assume the air density is \(1.3 \mathrm{kg} / \mathrm{m}^{3}\) both inside and outside. (b) What is the buoyant force on the beach ball in air? The thickness of the plastic is about 2 mm-negligible compared to the radius of the ball. (c) The ball is thrown straight up in the air. At the top of its trajectory, what is its acceleration? [Hint: When \(v=0,\) there is no drag force.]

Short answer

(a) The weight is approximately 1.41 N. (b) The buoyant force is around 0.427 N. (c) The acceleration at the top is about 6.85 m/s².

Step-by-step solution

Calculate Volume of the Beach Ball

Use the formula for the volume of a sphere, which is \( V = \frac{4}{3} \pi r^3 \), where \( r = 20.0 \ cm \) or \( 0.20 \ m \). This calculates to \( V = \frac{4}{3} \pi (0.20)^3 \approx 0.0335 \ m^3 \).

Calculate Mass of Air Inside

Mass of air is the product of its density and the volume it occupies. With air density \( \rho = 1.3 \ kg/m^3 \), the mass of the air inside is \( m_{air} = \rho \times V \approx 1.3 \times 0.0335 \approx 0.0435 \ kg \).

Calculate Total Mass of the Beach Ball

Add the mass of the plastic (\(0.10\ kg\)) and the mass of the air inside. The total mass \( m_{total} = 0.10 \ kg + 0.0435 \ kg = 0.1435 \ kg \).

Calculate Weight of the Beach Ball

The weight \( W \) of an object is given by \( W = m \cdot g \), where \( g = 9.81 \ m/s^2 \) is the acceleration due to gravity. Hence, \( W = 0.1435 \times 9.81 \approx 1.41 \ N \).

Calculate the Buoyant Force

The buoyant force \( F_b \) is given by the weight of the air displaced by the ball, \( F_b = \rho \cdot V \cdot g \). Thus, \( F_b = 1.3 \cdot 0.0335 \cdot 9.81 \approx 0.427 \ N \).

Determine Acceleration at Trajectory's Top

At the top of its trajectory, the only forces acting on the ball are its weight downward and buoyant force upward. The net force \( F_{net} = W - F_b \approx 1.41 - 0.427 = 0.983 \ N \). Using \( F = ma \), the acceleration \( a = \frac{F_{net}}{m_{total}} = \frac{0.983}{0.1435} \approx 6.85 \ m/s^2 \).

Key concepts

Buoyancy

Buoyancy is the force that allows objects to float or appear lighter when placed in a fluid, such as air or water. It acts in the opposite direction to gravity and is equal to the weight of the fluid displaced by the object. This principle was first described by the ancient Greek mathematician Archimedes. For the beach ball in this exercise, the buoyant force is the weight of the air that the ball displaces.

Understanding buoyancy helps in explaining how objects float. It can be calculated using the formula:

• \( F_b = \rho \cdot V \cdot g \)

where \( F_b \) is the buoyant force, \( \rho \) is the fluid density, \( V \) is the volume of the object, and \( g \) is the acceleration due to gravity.

Knowing the buoyant force is crucial for understanding the motion of the beach ball in air. As calculated, the force here equals approximately 0.427 N, which helps offset the weight of the ball slightly.

Force and Motion

In physics, force and motion are closely related concepts. A force causes an object to start moving, stop moving, or change direction. Newton's laws of motion describe how the forces acting on an object affect its motion. Particularly relevant to this problem is the second law, which states that the force applied to an object is equal to the mass of the object times its acceleration (\( F = ma \)).

For the beach ball, when it is thrown upwards, it initially moves against the force of gravity and the buoyant force. At the top of its trajectory, it temporarily stops due to gravity pulling it back. At this point, the net force acting on the ball is the difference between its weight and the buoyant force. Understanding these forces allows us to calculate the acceleration at that specific point, which is found to be approximately 6.85 m/s² downwards.

Air Density

Air density refers to the mass of air per unit volume and is a key factor in determining the buoyant force and the overall behavior of objects moving through air. It is usually measured in kilograms per cubic meter (kg/m³).

For this exercise, the air density is given as 1.3 kg/m³ both inside and outside the beach ball. This constant density allows us to make straightforward calculations for both the mass of the air inside the ball and the buoyant force. Because air density influences how much air is displaced by the ball, it directly connects to the amount of buoyant force experienced.

It's important to understand that variations in air density, such as those caused by changes in altitude or temperature, can affect these calculations. However, in this scenario, the density remains consistent, simplifying the problem.

Sphere Volume Calculation

Calculating the volume of a sphere is important in determining both the mass of the air inside and the displacement-related calculations for the buoyant force. The formula to find the volume of a sphere is:

• \( V = \frac{4}{3} \pi r^3 \)

where \( r \) is the radius of the sphere.

In this problem, the radius of the beach ball is given as 20 cm, or 0.20 m when converted to meters for use in the formula. Using the formula, we calculate the volume of the ball to be approximately 0.0335 cubic meters. This measure is crucial for determining both the mass of the air inside the ball and the buoyant force on the ball.

Ensuring accuracy in these calculations is essential, as errors could lead to incorrect assessments of the forces at play and the motion of the ball when thrown.