Work done in air resistance is the energy required to overcome the resistance force exerted by the air on an object as it moves through it. This force is dependent on the object’s speed, cross-sectional area, and shape, as well as the density of the air. The formula for calculating work done by air resistance is a crucial concept in physics, and understanding its intricacies can greatly benefit physics students.
The Formula for Work Done in Air Resistance
The formula for calculating work done by air resistance is:
W = ½ C A ρ v^3 t
Where:
- W is the work done (in joules, J)
- C is the drag coefficient (dimensionless)
- A is the cross-sectional area of the object (in square meters, m^2)
- ρ is the density of the air (in kilograms per cubic meter, kg/m^3)
- v is the velocity of the object (in meters per second, m/s)
- t is the time taken (in seconds, s)
Let’s break down each of these variables and understand their significance in the context of work done in air resistance.
Drag Coefficient (C)
The drag coefficient, C, is a dimensionless number that represents the amount of resistance an object experiences as it moves through a fluid, such as air. It is determined experimentally and depends on the object’s shape and orientation. Some common drag coefficients for various shapes are:
| Shape | Drag Coefficient (C) |
|---|---|
| Sphere | 0.47 |
| Cylinder (perpendicular to flow) | 1.2 |
| Flat plate (perpendicular to flow) | 1.98 |
| Streamlined body | 0.04 |
The drag coefficient is a crucial factor in determining the work done by air resistance, as it directly affects the magnitude of the resistance force.
Cross-Sectional Area (A)
The cross-sectional area, A, is the area of the object that is exposed to the air resistance. It is measured in square meters (m^2) and can be calculated using the formula:
A = π * r^2
Where r is the radius of the object.
The cross-sectional area of an object is directly proportional to the work done by air resistance, as a larger surface area will experience more resistance.
Air Density (ρ)
The density of the air, ρ, is measured in kilograms per cubic meter (kg/m^3) and can be affected by factors such as altitude and temperature. At sea level and 20°C, the air density is approximately 1.225 kg/m^3.
The air density is an important factor in the work done by air resistance, as a higher density will result in a greater resistance force.
Velocity (v)
The velocity of the object, v, is measured in meters per second (m/s) and is a crucial factor in the work done by air resistance. The formula shows that the work done is proportional to the cube of the velocity, meaning that even a small change in velocity can have a significant impact on the work done.
Time (t)
The time taken, t, is measured in seconds (s) and represents the duration over which the work is done. This factor is important when considering the total work done over a period of time, rather than just a single instant.
Calculating Work Done in Air Resistance: Examples
Let’s consider a few examples to illustrate the application of the work done in air resistance formula.
Example 1: Falling Coffee Filter
Suppose a physics student wants to measure the effect of air resistance on the motion of a coffee filter. They drop the filter from a height of 2 meters and measure the time it takes to reach the ground, which is 1.5 seconds.
Given:
– Cross-sectional area of the coffee filter (A) = 0.0025 m^2
– Drag coefficient of the coffee filter (C) = 1.1
– Air density (ρ) = 1.225 kg/m^3
– Time taken (t) = 1.5 s
– Velocity at impact (v) = 4.9 m/s (calculated using the formula v = sqrt(2gh), where g is the acceleration due to gravity)
Substituting the values into the work done by air resistance formula:
W = ½ * C * A * ρ * v^3 * t
W = ½ * 1.1 * 0.0025 m^2 * 1.225 kg/m^3 * (4.9 m/s)^3 * 1.5 s
W = 0.0375 J
This shows that the work done by air resistance on the falling coffee filter is 0.0375 joules.
Example 2: Projectile Motion
Consider a ball thrown upward with an initial velocity of 14 m/s, reaching a maximum height of 8.4 meters. Assume the ball has a mass of 0.37 kg.
To calculate the work done by air resistance, we need to find the final velocity of the ball as it returns to its starting point. We can use the principle of conservation of energy to do this.
At the maximum height, the total energy of the ball is equal to its initial potential energy plus the work done by air resistance:
Ep = m * g * h
Ep = 0.37 kg * 9.8 m/s^2 * 8.4 m
Ep = 30.5 J
At the starting point, the total energy of the ball is equal to its initial kinetic energy:
Ek = ½ * m * v^2
Ek = ½ * 0.37 kg * (14 m/s)^2
Ek = 72.1 J
Using the principle of conservation of energy, we can set the total energy at the maximum height equal to the total energy at the starting point:
Ep + W = Ek
30.5 J + W = 72.1 J
W = 72.1 J – 30.5 J
W = 41.6 J
However, the book’s answer is -5.8 J, which suggests that the air resistance is doing negative work on the ball, reducing its kinetic energy. This can be explained by the fact that the air resistance is acting in the opposite direction of the ball’s motion, reducing its speed and therefore its kinetic energy.
Experimental Investigations of Work Done in Air Resistance
In the context of physics students studying work done in air resistance, they may perform various experiments to measure the effect of air resistance on an object’s motion. Here are some examples of such experiments:
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Dropping Coffee Filters: Students can drop a single coffee filter and measure the time it takes to fall a certain distance, then repeat the experiment with a different number of filters to observe the effect of mass on the motion.
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Using a Motion Detector: Students can use a motion detector to measure the velocity of a falling object over time, allowing them to calculate the work done by air resistance.
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Inclined Plane Experiment: Students can roll an object down an inclined plane and measure the work done by air resistance by comparing the actual distance traveled to the theoretical distance calculated without air resistance.
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Parachute Experiment: Students can attach a parachute to an object and measure the work done by air resistance as the object falls, observing the effect of the parachute on the object’s motion.
These experiments not only help students understand the concept of work done in air resistance but also allow them to apply the theoretical knowledge to real-world situations.
Conclusion
Work done in air resistance is a fundamental concept in physics that is crucial for understanding the motion of objects through fluids. By understanding the formula for calculating work done by air resistance and the factors that influence it, physics students can gain a deeper understanding of the physical world around them and apply this knowledge to a wide range of problems and experiments.
References
- Is there another way to calculate the work done by air resistance? – Physics Stack Exchange
- Air Resistance on a Falling Object – Milligan Physics
- Calculating Air Resistance: How to Solve for Work Without Using Mass – Physics Forums
- Air resistance makes object go farther? – Reddit
- How Physics Includes Air Resistance in Calculations – YouTube
Hi, I’m Akshita Mapari. I have done M.Sc. in Physics. I have worked on projects like Numerical modeling of winds and waves during cyclone, Physics of toys and mechanized thrill machines in amusement park based on Classical Mechanics. I have pursued a course on Arduino and have accomplished some mini projects on Arduino UNO. I always like to explore new zones in the field of science. I personally believe that learning is more enthusiastic when learnt with creativity. Apart from this, I like to read, travel, strumming on guitar, identifying rocks and strata, photography and playing chess.