Calculating Drag Force on a Circular Disc
This question asks us to determine the force needed to hold a circular disc stationary against a wind. This force is equal in magnitude and opposite in direction to the drag force exerted by the wind on the disc.
The drag force ($F_D$) on an object in a fluid flow is calculated using the drag equation:
\( F_D = \frac{1}{2} \rho V^2 A C_D \)
Where:
- \( \rho \) is the density of the fluid (wind density in this case).
- \( V \) is the velocity of the fluid (wind speed).
- \( A \) is the reference area of the object normal to the flow.
- \( C_D \) is the coefficient of drag, which depends on the shape of the object and the flow conditions.
Given Parameters for Drag Force Calculation
We are given the following information:
- Diameter of the circular disc, \( D = 3 \) m
- Wind velocity, \( V = 26.4 \) m/s
- Wind density, \( \rho = 1.2 \) kg/m³
- Coefficient of drag of the disc, \( C_D = 1.1 \)
Determining the Reference Area
For a circular disc held normal to the wind, the reference area is the frontal area of the disc, which is the area of the circle. The area \( A \) of a circle with diameter \( D \) is given by:
\( A = \frac{\pi}{4} D^2 \)
Substituting the given diameter:
\( A = \frac{\pi}{4} (3 \text{ m})^2 \)
\( A = \frac{\pi}{4} \times 9 \text{ m}^2 \)
\( A \approx 7.0686 \text{ m}^2 \)
Calculating the Drag Force
Now we can substitute the values into the drag force formula:
\( F_D = \frac{1}{2} \rho V^2 A C_D \)
\( F_D = \frac{1}{2} \times (1.2 \text{ kg/m³}) \times (26.4 \text{ m/s})^2 \times (7.0686 \text{ m}^2) \times 1.1 \)
\( F_D = 0.6 \text{ kg/m³} \times (696.96 \text{ m²/s²}) \times 7.0686 \text{ m²} \times 1.1 \)
\( F_D = 0.6 \times 696.96 \times 7.0686 \times 1.1 \text{ N} \)
\( F_D \approx 3246.7 \text{ N} \)
Converting Force to KiloNewtons
The options are given in kiloNewtons (kN). To convert Newtons (N) to kiloNewtons (kN), we divide by 1000:
\( F_D \approx \frac{3246.7 \text{ N}}{1000} \)
\( F_D \approx 3.2467 \text{ kN} \)
Comparing with Options
The calculated drag force is approximately 3.2467 kN. Let's compare this value with the given options:
| Option | Value |
|---|---|
| 1 | 1.25 kN |
| 2 | 2.5 kN |
| 3 | 3.25 kN |
| 4 | 4.2 kN |
The calculated value of 3.2467 kN is very close to 3.25 kN.
Summary of Drag Force Calculation
The force required to hold the circular disc at rest against the wind is equal to the drag force. Using the given parameters and the drag force formula, we calculated the force to be approximately 3.2467 kN.
Revision Table: Key Concepts
| Concept | Description |
|---|---|
| Drag Force | The resistance force exerted by a fluid on an object moving through it, or on an object held stationary in a moving fluid. |
| Drag Coefficient (\( C_D \)) | A dimensionless quantity that quantifies the drag or resistance of an object in a fluid environment. It depends on the shape and surface characteristics of the object. |
| Reference Area (\( A \)) | The area of the object used in the drag equation, typically the frontal area projected onto a plane perpendicular to the flow direction. |
| Fluid Density (\( \rho \)) | A measure of mass per unit volume of the fluid. Denser fluids exert greater drag. |
| Fluid Velocity (\( V \)) | The speed of the fluid relative to the object. Drag force is proportional to the square of the velocity. |
Additional Information on Aerodynamic Drag
Aerodynamic drag is a type of fluid friction, or fluid resistance, acting on an object due to motion through a fluid (like air or water). It is composed of several components:
- Form Drag (Pressure Drag): Caused by the shape of the object. Bluff or non-streamlined shapes create larger pressure differences between the front and back, leading to higher form drag. A circular disc has high form drag when facing the wind.
- Skin Friction Drag: Caused by the friction between the fluid and the surface of the object. This is more significant for streamlined shapes or rough surfaces.
- Interference Drag: Occurs when multiple parts of an object interact, affecting the airflow around each other (less relevant for a simple disc).
- Lift-Induced Drag: Relevant for objects generating lift (like wings), not applicable to a flat disc held normal to the flow.
The coefficient of drag ($C_D$) is an empirical value determined through experiments. It lumps together the effects of form drag and skin friction drag for a specific object shape under particular flow conditions (often represented by the Reynolds number). For a flat disc normal to the flow, the $C_D$ value is relatively high, reflecting the significant pressure difference created across its front and back surfaces.
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