Problem 42 - College Physics

Question

An atmospheric drag force with magnitude \(F_{D}=D v^{2},\) where \(v\) is speed, acts on a falling \(300 \mathrm{mg}\) raindrop that reaches a terminal velocity of \(11 \mathrm{~m} / \mathrm{s}\). (a) Show that the \(\mathrm{SI}\) units of \(D\) are \(\mathrm{kg} / \mathrm{m}\). (b) Find the value of \(D .\)

Step-by-Step Solution

Answer

(a) The SI units of \(D\) are \(\mathrm{kg/m}\). (b) The value of \(D\) is \(2.43 \times 10^{-5}\, \mathrm{kg/m}.\)

Step 1: Identify SI Units for Each Variable

To determine the SI units of the drag coefficient \(D\), start by identifying the SI units of each variable in the drag force equation. The force \(F_D\) is given in newtons \((N)\), which has SI units of \(\text{kg} \cdot \text{m/s}^2\). The velocity \(v\) is in meters per second \((m/s)\).

Step 2: Rearrange the Equation for D

The drag force equation is given by \(F_D = D v^2\). We need to isolate \(D\), so rearrange the equation to solve for \(D\):\[ D = \frac{F_D}{v^2} \]

Step 3: Substitute Units into the Rearranged Equation

Substitute the units into the rearranged equation:\[ \text{units of } D = \frac{\text{units of } F_D}{\text{units of } v^2} = \frac{\text{kg} \cdot \text{m/s}^2}{(\text{m/s})^2} \]

Step 4: Simplify the Units

Simplify the units expression by canceling the \((m/s)^2\) terms:\[ \text{units of } D = \frac{\text{kg} \cdot \text{m/s}^2}{\text{m}^2/\text{s}^2} = \frac{\text{kg}}{\text{m}} \]

Step 5: Set Up Equation for D Using Given Values

To find the value of \(D\), use the equation \(D = \frac{F_D}{v^2}\). At terminal velocity, the force due to gravity equals the drag force. The force of gravity on the raindrop is given by:\[ F_g = m \cdot g = 0.0003 \text{ kg} \cdot 9.8 \text{ m/s}^2 = 0.00294 \text{ N} \]

Step 6: Calculate the Value of D

Substitute the known values into the equation for \(D\):\[ D = \frac{0.00294 \text{ N}}{(11 \text{ m/s})^2} = \frac{0.00294 \text{ N}}{121 \text{ m}^2/\text{s}^2} = 2.43 \times 10^{-5} \text{ kg/m} \]

Key Concepts

Drag Coefficient

The drag coefficient, symbolized as "D" in the context of the exercise, quantifies how aerodynamic a body is as it moves through a fluid, such as air. It is a dimensionless number that represents the resistance faced by an object when moving through a fluid. This resistance is called drag force.

Drag force is an essential aspect of study because it affects the acceleration and speed of an object falling through the atmosphere. The drag coefficient helps us understand how different shapes and textures can influence this force. For example, a streamlined shape like a raindrop will have a lower drag coefficient compared to a flat plate pushed through the air.

The formula for drag force is given by:

• \( F_D = D \cdot v^2 \)
• Where \( F_D \) is the drag force and \( v \) is the speed or velocity of the object.

Thus, the drag coefficient "D" is found by rearranging this formula, indicating how efficiently the velocity of the object and the force applied relate to each other. Understanding this helps us design better vehicles, sports equipment, and even clothing that can benefit from reduced air resistance.

Terminal Velocity

Terminal velocity is a fascinating concept that describes the constant speed that a freely falling object eventually reaches when the resistance of the medium (air, in most cases) prevents further acceleration. At this speed, the force of gravity downward is balanced by the drag force upward.

This means the object will no longer continue to speed up and will fall at a steady rate. For our raindrop example in the exercise, it reaches a terminal velocity of 11 meters per second, meaning it is falling through the air at a speed where the downward pull of gravity matches the upward drag force.

When understanding terminal velocity, consider these key points:

• The velocity at which the drag force equals the force of gravity on the object.
• The object's speed remains constant once reached.
• It depends on factors like the object's size, shape, and mass, as well as the density of the medium it falls through.

Researchers study terminal velocity to optimize designs that either prevent it (like parachutes that slow descent) or take advantage of it (like streamlining vehicles for steady speeds without excess fuel consumption).

SI Units

Units of measurement are fundamental to scientific studies, and the International System of Units (SI) provides a standard for expressing physical quantities. Concerning the exercise, understanding SI units is crucial for accurately calculating the drag coefficient "D" and interpreting the results of our solution.

In the context of the exercise, these units are used:

• Force \(F_D\) is measured in newtons (N), where 1 newton is equivalent to \( \text{kg} \cdot \text{m/s}^2 \).
• Velocity \(v\) is measured in meters per second (m/s).

By substituting these units into the equation, we find that the units for the drag coefficient "D" simplify to \( \text{kg/m} \). Understanding and using SI units allows scientists and students across the globe to communicate findings and calculations accurately and universally. This makes experiments repeatable and results verifiable by the scientific community at large.

Force of Gravity

The force of gravity is a natural phenomenon by which all things with mass or energy are attracted towards one another. On Earth, gravity gives weight to physical objects and causes the skydiving effect experienced by falling bodies. In physics, the force of gravity is often simplified to "g," the acceleration caused by this force, approximately \(9.8 \, \text{m/s}^2\) on Earth's surface.

In our earlier example, calculating the gravitational force experienced by the raindrop involves using the formula:

• \( F_g = m \cdot g \)
• where \( m \) is mass in kilograms and \( g \) is the acceleration due to gravity.

Using this formula, we found that the small 300 mg raindrop experiences a gravitational force of approximately 0.00294 newtons. This force is crucial in determining terminal velocity, as it ultimately equals the drag force at this point, allowing the object to fall at a constant speed. Understanding gravity is key to unlocking knowledge about motion and forces both on Earth and across the cosmos.