A hummingbird is able to hover because, as the wings move downwards, they exert a downward force on the air. Newton's third law tells us that the air exerts an equal and opposite force (upwards) on the wings. The average of this force must be equal to the weight of the bird when it hovers. If the wings move through a distance of 3.1 cm with each stroke, and the wings beat 96 times per second, determine the work performed by the wings on the air in 1 minute if the mass of the hummingbird is 3.0 grams.
To determine the work performed by the wings on the air in 1 minute, let's break down the problem into smaller steps.
Step 1: Calculate the distance traveled by the wings in 1 second:
Since the wings move through a distance of 3.1 cm with each stroke, and the wings beat 96 times per second, we can multiply these values to find the distance traveled by the wings in 1 second:
Distance traveled by the wings in 1 second = 3.1 cm/stroke * 96 strokes/second
Step 2: Convert the distance to meters:
We need to convert the distance from centimeters to meters to maintain consistent units:
Distance traveled by the wings in 1 second = (3.1 cm/stroke * 96 strokes/second) / 100 cm/m
Step 3: Calculate the force exerted by the wings on the air:
Since the wings exert a downward force on the air, and Newton's third law states that the air exerts an equal and opposite force (upwards) on the wings, we know that the average of these forces equals the weight of the bird when it hovers. Therefore, the force exerted by the wings on the air is equal to the weight of the bird:
Force exerted by the wings = Weight of the bird = mass of the bird * acceleration due to gravity
Given that the mass of the hummingbird is 3.0 grams, we need to convert it to kilograms to match the units of force (Newtons):
Weight of the bird = 3.0 grams * (1 kg / 1000 grams) * (9.8 m/s^2)
Step 4: Calculate the work performed by the wings on the air in 1 second:
Work is defined as force multiplied by distance. Using the previously determined values for force and distance:
Work done by the wings in 1 second = Force exerted by the wings * Distance traveled by the wings in 1 second
Step 5: Convert the work done to minutes:
To calculate the work done by the wings on the air in 1 minute, we need to convert the work done in 1 second to work done in 1 minute:
Work done by the wings in 1 minute = Work done by the wings in 1 second * 60 seconds/minute
By following these steps and plugging in the appropriate values, you can calculate the work performed by the wings on the air in 1 minute.