For a small spherical object falling through a fluid, its terminal velocity is directly proportional to the square of its radius.
Understanding Terminal Velocity
Terminal velocity is the maximum constant speed that an object reaches when falling through a fluid (like air or water). This occurs when the downward force of gravity is perfectly balanced by the upward forces of fluid resistance (drag) and buoyancy. At this point, the net force on the object is zero, and it stops accelerating, continuing to fall at a steady speed.
Several factors influence an object's terminal velocity, including its mass, shape, and size, as well as the properties of the fluid it's falling through, such as its density and viscosity.
How Radius Affects Terminal Velocity
For small spherical objects, the relationship between terminal velocity and radius is quite specific due to the nature of the forces acting upon them. The terminal velocity ($v$) of a small spherical object is directly proportional to the square of its radius ($R$). This means that if you double the radius of such an object, its terminal velocity will increase by a factor of four ($2^2$).
Let's break down why this relationship holds true:
- Gravitational Force: The force pulling the object down is its weight, which depends on its mass. For a spherical object, mass is proportional to its volume, and volume is proportional to the cube of its radius ($R^3$). So, gravitational force $\propto R^3$.
- Buoyant Force: This upward force is equal to the weight of the fluid displaced by the object. Like gravity, it also depends on the object's volume, so buoyant force $\propto R^3$.
- Drag Force (Fluid Resistance): For small spherical objects moving at relatively slow speeds through a viscous fluid, the drag force is described by Stokes' Law. According to Stokes' Law, the drag force is directly proportional to the object's radius ($R$), its velocity ($v$), and the fluid's viscosity. So, drag force $\propto R \times v$.
At terminal velocity, the net force is zero:
(Gravitational Force - Buoyant Force) = Drag Force
Since both gravitational and buoyant forces are proportional to $R^3$, their difference is also proportional to $R^3$.
Therefore: $R^3 \propto R \times v$
To find the relationship for $v$, we can rearrange this:
$v \propto R^3 / R$
$v \propto R^2$
This fundamental relationship, derived from the balance of these forces, clearly demonstrates that terminal velocity scales with the square of the radius for small spherical objects.
Practical Implications and Examples
Understanding this relationship has significant practical implications in various fields:
- Sedimentation: Smaller particles settle much more slowly than larger ones. For instance, tiny dust particles can remain suspended in the air for extended periods, while larger raindrops fall quickly. This principle is crucial in processes like water treatment (settling tanks) and geology (sediment analysis).
- Particle Separation: In industrial and scientific applications, techniques like centrifugation leverage this principle to separate particles of different sizes or densities. Larger particles reach terminal velocity faster and settle at the bottom more quickly.
- Atmospheric Science: The fall rates of aerosols, cloud droplets, and precipitation are governed by this dependency, influencing weather patterns and air quality.
| Radius (R) | Terminal Velocity (v) (Arbitrary Units) | Observation |
|---|---|---|
| 1 unit | 1 unit | Baseline |
| 2 units | 4 units ($2^2$) | Quadruples |
| 3 units | 9 units ($3^2$) | Increases ninefold |
| 4 units | 16 units ($4^2$) | Increases sixteenfold |
As illustrated, a slight increase in radius leads to a disproportionately larger increase in terminal velocity, making size a critical factor in how quickly small objects fall through fluids.
Factors Beyond Radius
While radius is a major determinant for small spherical objects, other factors also play a crucial role in terminal velocity:
- Object Density: Denser objects (with more mass for their size) experience a greater gravitational pull and thus achieve a higher terminal velocity.
- Fluid Density: A denser fluid provides more buoyant force, which reduces the net downward force and therefore lowers the terminal velocity.
- Fluid Viscosity: A more viscous fluid creates greater drag. This increased resistance slows the object down, resulting in a lower terminal velocity.
- Object Shape: Although this question specifies spherical objects, it's worth noting that non-spherical objects have different drag characteristics due to their varied shapes, which significantly affects their terminal velocity.
In conclusion, for small spherical objects, the terminal velocity is directly proportional to the square of the radius. This fundamental physical relationship is critical for understanding the motion of particles in fluids, from microscopic biological cells to atmospheric aerosols.
For further reading on terminal velocity and Stokes' Law, you can refer to educational resources on fluid dynamics or physics of falling objects.
