Question

An aluminum sphere (specific gravity \(=2.7\) ) falling through water reaches a terminal speed of \(5.0 \mathrm{cm} / \mathrm{s}\) What is the terminal speed of an air bubble of the same radius rising through water? Assume viscous drag in both cases and ignore the possibility of changes in size or shape of the air bubble; the temperature is \(20^{\circ} \mathrm{C}\)

Short answer

The terminal speed of the air bubble is approximately 2.94 cm/s.

Step-by-step solution

Understanding the Problem

We're asked to find the terminal speed of an air bubble with the same radius as an aluminum sphere in water. The terminal speed for the sphere is given as 5.0 cm/s. The problem involves comparing the dynamics of two spheres affected by gravity and buoyancy in water.

Calculate Effective Density for the Sphere

Determine the force due to gravity for the aluminum sphere using its specific gravity. Specific gravity implies the ratio of the density of the sphere compared to water, so the effective density \( \rho_s \) is 2.7 times that of water's density \( \rho_w \). Thus, \( \rho_s = 2.7 \rho_w \).

Determine Buoyant Force and Drag Force for Sphere

The aluminum sphere reaches terminal velocity when the force of gravity (weight minus buoyant force) is opposed by the fluid's drag force. This is characterized by the balance: \( \frac{4}{3} \pi r^3 (\rho_s - \rho_w) g = 6 \pi \eta r v_s \). Here, \( \eta \) is the dynamic viscosity of water, \( v_s = 5.0 \text{ cm/s} \) is the terminal speed, and \( r \) is the radius of the sphere.

Calculate Terminal Velocity of Air Bubble

An air bubble rising through water also experiences forces of buoyancy and drag, but in this case, the effective density \( \rho_b \) of the bubble is much lower than that of water. Given the balance: \( \frac{4}{3} \pi r^3 (\rho_w - \rho_b) g = 6 \pi \eta r v_b \), for the bubble, since \( \rho_b \approx 0 \), it simplifies to \( \frac{4}{3} \pi r^3 \rho_w g = 6 \pi \eta r v_b \).

Formulate Ratio for Velocity

From the equilibrium conditions for both the sphere and the bubble, the ratio of terminal velocities \( v_b/v_s \) is given by \( (\rho_w - \rho_b) / (\rho_s - \rho_w) \). With \( \rho_b \approx 0 \), this simplifies to \( v_b/v_s = \rho_w / (2.7 \rho_w - \rho_w) = 1.0 / (2.7 - 1) = 1/1.7 \).

Solve for Terminal Velocity of the Bubble

Using the velocity ratio, we have \( v_b = (1/1.7) \times 5.0 \cm/s \approx 2.94 \cm/s \). Thus, the terminal speed of the air bubble rising through water is approximately 2.94 cm/s.

Key concepts

Viscous Drag

When an object moves through a fluid, such as water or air, it experiences a force that opposes its motion called viscous drag. This drag force is caused by the interaction between the object's surface and the fluid molecules. Unlike other forms of resistance, viscous drag is heavily dependent on the velocity of the object. As you move faster, the drag force increases proportionally, making it a key factor in reaching terminal velocity.

For spheres, the Stokes' Law formula is used to calculate this drag: \( F_d = 6 \pi \eta r v \), where \( \eta \) is the dynamic viscosity of the fluid, \( r \) is the radius of the sphere, and \( v \) is the velocity of the object. This equation highlights how both the viscosity of the fluid and the size of the object impact the drag force experienced.

Buoyancy

Buoyancy is the upward force exerted by a fluid on an object that is partly or completely submerged. This force occurs because the pressure in a fluid increases with depth, pushing up against the object. The buoyant force equals the weight of the fluid displaced by the object, as described by Archimedes' principle. If the buoyant force equals the object's weight, it will float; if less, it will sink.

In the context of terminal velocity, buoyancy offsets some of the gravitational force acting on the object, reducing the effective downward force. A lighter object like an air bubble experiences greater buoyancy compared to the dense aluminum sphere, affecting their respective terminal velocities.

Specific Gravity

Specific gravity is a dimensionless number that indicates how dense a substance is compared to water. It’s calculated by dividing the density of the object by the density of water. For instance, aluminum has a specific gravity of 2.7, meaning it is 2.7 times denser than water. Since water has a specific gravity of 1, this comparison becomes straightforward.

Understanding specific gravity helps predict how an object will behave when submerged. High specific gravity indicates that the object is denser and likely to sink, whereas lower specific gravity (like air's ~0) indicates a tendency to rise in a fluid.

Density

Density is a physical property that measures how much mass is contained in a given volume. It is typically expressed in units such as kilograms per cubic meter (kg/m³). The density of an object or fluid can influence many properties, such as buoyancy and stability. In equations, it is often denoted by \( \rho \).

In fluid dynamics, the difference in density between an object and the fluid it is in determines the object's buoyant force and eventual movement. A denser object may sink, while one with lower density, such as a bubble, will rise. Thus, density directly impacts the effective forces acting on an object, influencing its terminal speed in a fluid.

Dynamic Viscosity

Dynamic viscosity is a measure of a fluid's resistance to flow. It helps describe how "thick" or "sticky" a fluid is, with higher values indicating a greater resistance to flow. Viscosity plays a pivotal role in determining the viscous drag an object faces when moving through the fluid. It is denoted as \( \eta \) and typically measured in units such as Pascal-seconds (Pa·s).

In the context of the exercise, water has a specific dynamic viscosity value, which affects how both the aluminum sphere and air bubble move through water. Since viscous drag is dependent on this property, understanding dynamics viscosity helps predict and calculate terminal velocities in various fluids.