Problem 73 - College Physics

Question

\(\bullet\) \(\bullet\) A typical flying insect applies an average force equal to twice its weight during each downward stroke while hovering. Take the mass of the insect to be \(10 \mathrm{g},\) and assume the wings move an average downward distance of 1.0 \(\mathrm{cm}\) during each stroke. Assuming 100 downward strokes per second, estimate the average power output of the insect.

Step-by-Step Solution

Answer

The average power output of the insect is approximately 0.196 W.

Step 1: Understanding the Problem

The problem is about estimating the average power output of an insect while hovering. The insect applies a force twice its weight during each downward stroke. Our goal is to find how much work is done per second and thus the power output.

Step 2: Calculate the Weight of the Insect

First, we calculate the weight of the insect. The mass is given as \(10\, \text{g} = 0.01\, \text{kg}\). The force due to gravity (weight) is given by \( F_g = mg \), where \( g = 9.8\, \text{m/s}^2 \).\[ F_g = 0.01 \times 9.8 = 0.098\, \text{N} \]

Step 3: Determine the Force Applied

The insect applies a force twice its weight during each stroke, so the force applied is:\[ F = 2 \times 0.098 = 0.196 \, \text{N} \]

Step 4: Calculate the Work Done per Stroke

The work done in one stroke is the force applied multiplied by the distance moved. The average downward distance is given as 1.0 cm, which is \(0.01\, \text{m}\).\[ W = F \times d = 0.196 \times 0.01 = 0.00196 \, \text{J} \]

Step 5: Calculate the Power Output

Power is the work done per second. There are 100 strokes per second, so the total work done in one second is:\[ ext{Total Work per second} = 0.00196 \, \text{J/stroke} \times 100 \, \text{strokes/s} = 0.196 \, \text{J/s} \]Thus, the power output (which is work done per unit time) is 0.196 W.

Key Concepts

Work Done

When we talk about the concept of "work done" in physics, we specially focus on how force causes displacement. In simple terms, work is done when an object is moved by a force over a distance. In the context of our hovering insect, each downward stroke of the wings involves work because a force is exerted to move the wings downwards.
If we go back to the formula for work done, it is given by:

• Work Done (W) = Force (F) × Distance (d)

For our insect, the force applied is twice its own weight, and the distance is how far the wings move down with each stroke. The calculation starts by determining the force, and then it is simply multiplied by the distance traveled per stroke to find the work done. Keep in mind, in this problem, the work is done during every downward movement of the wings, making this an ongoing cycle as the insect hovers.

Force Calculation

Force calculation is a fundamental part of physics, especially in problems dealing with motion and work, just like with our hovering insect. Calculating force here involves understanding weight and how it relates to the gravitational pull on an object.
For a simple start, the weight of an insect, just like any object, can be determined by:

• Force due to gravity (weight, Fg) = Mass (m) × Gravitational acceleration (g)

Where gravity is approximately 9.8 m/s². The exercise specifies that the insect applies a force twice its weight during each wing stroke. Hence, once you determine the weight of the insect, you multiply it by two to find the total force exerted per stroke. This gives a clear insight into the physical effort the insect puts forth to stay aloft.

Hovering Insects

Hovering insects are a fascinating example of nature's engineering and physics in action. For an insect to hover, it must exert enough force with its wings to counteract gravitational forces pulling it down.
This action of hovering is not just about moving up and down; it involves maintaining a delicate balance between lift and gravity, achieved by the rapid movement of their wings. The faster the wings move, the more they can resist gravity and keep the insect airborne. In our problem, we assume 100 downward strokes per second, which is a significant number of repetitions to sustain hovering.
The concept of hovering may seem simple, but it involves a complex interplay of forces and energy transfer, making insects exceptional at regulating energy to remain suspended in the air with minimal energy waste.

Energy per Stroke

The notion of "energy per stroke" focuses on how much energy an insect spends with each wing movement. In this context, energy and work done per stroke are closely linked because, in physics, energy is essentially the capacity to do work.
The total energy used by the insect per second in hovering is the sum from all its wing strokes. Each stroke expends a specific amount of work, calculated using the formula:

• Work Done = Force applied × Distance per stroke

This method calculates the energy each stroke consumes. With our example, since the insect flaps its wings at a rate of 100 strokes per second, understanding the energy used per stroke helps calculate the total energy consumption or power output. By multiplying the work done per stroke by the number of strokes per second, you gain insight into how this activity translates into the insect's power output. This provides a quantitative grasp of how dynamic and energetically intensive the process is for seemingly tiny animals to continue hovering.