Problem 61 - University Physics with Modern Physics

Question

While hovering, a typical flying insect applies an average force equal to twice its weight during each downward stroke. Take the mass of the insect to be 10 g, and assume the wings move an average downward distance of 1.0 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 0.196 W.

Step 1: Calculate the Weight of the Insect

First, we need to find the force due to gravity, acting on the insect. Given the mass of the insect is 10 g, convert this mass into kilograms to use the standard unit: \[ m = 10 \text{ g} = 0.01 \text{ kg} \]The gravitational acceleration \( g \) is approximately \( 9.8 \text{ m/s}^2 \).The weight \( W \) of the insect is given by:\[ W = m \cdot g = 0.01 \text{ kg} \times 9.8 \text{ m/s}^2 = 0.098 \text{ N} \]

Step 2: Determine the Force Per Stroke

The average force applied by the insect during each downward stroke is twice its weight. Therefore, we calculate this force \( F_{stroke} \) as:\[ F_{stroke} = 2 \times W = 2 \times 0.098 \text{ N} = 0.196 \text{ N} \]

Step 3: Compute the Work Done Per Stroke

Work done per stroke \( W_{stroke} \) is calculated by multiplying the force per stroke by the distance the wings move downward. Convert the distance from cm to meters for the calculation:\[ d = 1 \text{ cm} = 0.01 \text{ m} \]So,\[ W_{stroke} = F_{stroke} \times d = 0.196 \text{ N} \times 0.01 \text{ m} = 0.00196 \text{ J} \]

Step 4: Calculate the Power Output

Power is the rate at which work is done. The power output \( P \) can be calculated by multiplying the work done per stroke by the number of strokes per second:\[ f = 100 \text{ strokes/second} \]\[ P = W_{stroke} \times f = 0.00196 \text{ J} \times 100 = 0.196 \text{ W} \]

Key Concepts

Force Calculation

Insect flight involves interesting physics, especially when it comes to calculating the forces that enable hovering. Force calculation in this context is crucial for understanding how an insect maintains flight. Usually, insects exert force by flapping their wings. This force is often greater than their weight to achieve lift.
Let's consider our exercise: the insect applies a force twice its weight during each downward stroke. By knowing the insect's mass, we can calculate the gravitational force, commonly known as weight.To break it down:

• First, we convert the mass from grams to kilograms: 10 g = 0.01 kg.
• Then, by using the acceleration due to gravity (9.8 m/s²), we calculate the weight: \[ W = m \times g = 0.01 \text{ kg} \times 9.8 \text{ m/s}^2 = 0.098 \text{ N} \]

Here, weight acts as the downward force due to gravity. The insect needs to apply a force greater than this to raise itself upward, which, as given, is twice the weight. So, calculate:

• The force per stroke: \[ F_{stroke} = 2 \times W = 0.196 \text{ N} \]

This explains how insects lift themselves by creating sufficient upward force through wing flapping.

Work and Energy

When an insect flaps its wings, it performs work. Work in physics relates to the amount of energy transferred by a force over a distance. In this exercise, the work done during each stroke involves the thrust created by the wings moving through the air.
Let's calculate the work done per wing stroke:

• Distance the wings move downward is given as 1 cm. In meters, that's 0.01 m.
• Use the formula for work done: \[ W_{stroke} = F_{stroke} \times d = 0.196 \text{ N} \times 0.01 \text{ m} = 0.00196 \text{ J} \]

The calculation tells us how much energy is transferred in each stroke to both overcome gravitational forces and sustain flight. Work is essentially the energy needed to move an object over a certain distance. Here, it measures the energy required for the wings' downward movement.

Power in Physics

Power in physics refers to how quickly work is done or energy is transferred. In our flying insect scenario, power tells us about the speed of energy usage during flight. Calculating power helps us understand efficiency and endurance of the insect in flight.
Power is calculated as work done per unit time, represented as:

• Considering 100 strokes per second (frequency), we calculate the power output: \[ P = W_{stroke} \times f = 0.00196 \text{ J} \times 100 \text{ strokes/second} = 0.196 \text{ W} \]

This 0.196 W indicates how much energy the insect produces each second to maintain hovering. It highlights the remarkable energy efficiency of insects, despite their small size, and the rapidity with which their wings operate. Understanding these calculations offers insights into the power dynamics critical for sustaining insect flight.