Question

Two wooden balls of equal volume but different density are held beneath the surface of a container of water. Ball A has a density of \(0.5 \frac{\mathrm{g}}{\mathrm{m}^{3}},\) and ball \(\mathrm{B}\) has a density of \(0.7 \frac{\mathrm{g}}{\mathrm{cm}^{3}} .\) When the balls are released, they will accelerate upward to the surface. What is the relationship between the acceleration of ball A and that of ball B? A. Ball A has the greater acceleration. B. Ball B has the greater acceleration. C. Balls A and B have the same acceleration. D. It cannot be determined from information given.

Short answer

A. Ball A has the greater acceleration.

Step-by-step solution

- Understand the Situation

We have two balls of equal volume but different densities submerged in water. We need to determine the relationship between their accelerations when they are released.

- Recall the Buoyant Force

The buoyant force acting on an object submerged in a fluid is given by Archimedes' principle, which states that the buoyant force is equal to the weight of the fluid displaced by the object. Since the volumes of balls A and B are equal, the buoyant force on each ball will be the same.

- Calculate the Net Force

The net upward force on each ball will be the buoyant force minus the weight of the ball. Weight of a ball can be calculated using the formula: Weight = Density x Volume x Gravitational acceleration (g).

- Express the Accelerations

According to Newton's second law, the acceleration of each ball can be given by: \[ a = \frac{F_{net}}{m} \]Where, \( F_{net} \) is the net force and \( m \) is the mass of the ball. Since mass \( m \) is Density \( \times \) Volume, higher density implies higher mass for the same volume.

- Compare the Densities

Given that ball A has a density of 0.5 g/cm^3 and ball B has a density of 0.7 g/cm^3, ball A has a lesser mass compared to ball B given the same volume.

- Determine the Relationship

Since the buoyant force is the same for both, the ball with the lesser mass (ball A) will experience a greater acceleration upwards because of its smaller inertia.

Key concepts

Buoyant Force

The buoyant force is a key concept in fluid dynamics. It is the upward force exerted by a fluid on a submerged object. According to Archimedes' principle, this force is equal to the weight of the fluid that the object displaces.

In mathematical terms, the buoyant force \( F_{b} \) can be expressed as:

\[ F_{b} = \rho_{fluid} \times V_{object} \times g \]

where \( \rho_{fluid} \) is the density of the fluid, \( V_{object} \) is the volume of the object, and \( g \) is the acceleration due to gravity.

Since the volumes of balls A and B in the problem are equal, they displace the same amount of water, providing each ball with the same buoyant force.

Archimedes' Principle

Archimedes' principle is a fundamental tenet in fluid mechanics that explains how buoyant forces work. It states that any object wholly or partially submerged in a fluid experiences an upward force equal to the weight of the fluid it displaces.

For instance, when you place a ball in water, it pushes away some of the water to make room for itself. The water pushes back with a force equal to the weight of the pushed-away water. This upward push is what we call the buoyant force.

This principle is crucial in understanding why objects float or sink and is directly applied in our problem to understand the forces acting on balls A and B.

Acceleration Due to Buoyancy

Acceleration due to buoyancy occurs when the upward buoyant force acting on an object is greater than the downward gravitational force, causing the object to rise. To find the acceleration of the object, we use Newton's second law of motion:

\[ a = \frac{F_{net}}{m} \]

In our problem, the net force \( F_{net} \) is the buoyant force minus the weight of the ball:

\[ F_{net} = F_{b} - \text{Weight of ball} \]

Since both balls have the same buoyant force but different weights due to their different densities, the ball with the lesser mass (ball A) will have a greater acceleration. This is because the net force divided by a smaller mass (ball A) results in greater acceleration.

Density and Mass Relationship

Density is defined as mass per unit volume. It is a measure of how much matter an object contains in a given space. The formula for density \( \rho \) is:

\[ \rho = \frac{m}{V} \]

where \( m \) is the mass and \( V \) is the volume. From this equation, we can derive that mass \( m \) is:

\[ m = \rho \times V \]

In our problem, both balls have the same volume but different densities. Ball A has a density of 0.5 g/cm³ and ball B has a density of 0.7 g/cm³. Therefore, ball A has a lesser mass than ball B for the same volume.

Because mass is directly related to density times volume, a lower density implies a lower mass for an object of the same volume. Consequently, a smaller mass means lesser inertia, and when buoyant forces are equal, the ball with smaller mass (ball A) will accelerate more.