Question

What is the power output of a 1.4-g fly as it walks straight up a windowpane at \(2.3 \mathrm{~cm} / \mathrm{s}\) ?

Short answer

The power output is approximately 0.31 milliwatts.

Step-by-step solution

Understand the Problem

We need to find the power output of a fly walking straight up a windowpane. Key information given includes the mass of the fly (1.4 grams) and its vertical speed (2.3 cm/s). Power in this context is the rate at which the fly does work against gravity to move vertically.

Convert Units

First, convert the mass of the fly into kilograms because the standard unit for mass in physics is kilograms. We have 1.4 grams, which is 0.0014 kg, since 1 gram = 0.001 kg. The speed is given in cm/s, which has the standard meter unit converted as follows: 2.3 cm/s = 0.023 m/s.

Calculate Gravitational Work

The work done against gravity when the fly moves vertically is given by the formula: work = force × distance. The force in this case is the gravitational force, which has a magnitude of \( F = mg \), where \( m = 0.0014 \text{ kg} \) and \( g = 9.8 \text{ m/s}^2 \) (acceleration due to gravity).

Calculate Power Output

Power (\( P \)) is the rate of doing work or the work done per unit of time. Since power is also the product of force and velocity when force is in the direction of velocity, we can use \( P = Fv \):\[ P = (0.0014 \text{ kg} \times 9.8 \text{ m/s}^2) \times 0.023 \text{ m/s} \]Calculate to find \( P \).

Perform the Calculation

Calculate the force using the mass and gravity: \( F = 0.0014 \text{ kg} \times 9.8 \text{ m/s}^2 = 0.01372 \text{ N} \).Then, find the power using this force and the velocity: \( P = 0.01372 \text{ N} \times 0.023 \text{ m/s} = 0.00031556 \text{ W} \). Hence, the power output is approximately 0.31 milliwatts.

Key concepts

Gravitational Work

Gravitational work refers to the work done by a force in the direction of gravitational acceleration. In the context of a fly walking up a windowpane, this work is required to overcome the gravitational force pulling the fly downward. The gravitational force is determined by the fly's weight, which depends on its mass and the gravitational acceleration constant, denoted as \( g \).

Calculating the work done against gravity involves understanding the formula: \[ \text{Work} = \text{Force} \times \text{Distance} \] For vertical motion, the force involved is the gravitational force \( (mg) \), and distance would be equivalent to how far the fly travels vertically. However, in this exercise, we first use force and velocity to find power. It's important to keep units in standard forms so that they correctly communicate the quantities we measure. Here, mass is in kilograms, and distance (implicitly used by velocity) is in meters. By mastering the concept of gravitational work, it's easier to understand how organisms or objects require energy to move against gravity, and why factors like mass and velocity play critical roles.

Unit Conversion

Unit conversion is a crucial skill in physics that ensures consistency and accuracy during calculations. To match the standard units used in equations, we often need to convert the given units into SI units.

In this problem, the mass of the fly is given in grams, but the standard unit of mass in physics is kilograms. Therefore, conversion is simple: 1 gram equals 0.001 kilograms. Thus, 1.4 grams is converted to 0.0014 kg.
For velocity, given in centimeters per second, the standard unit should be meters per second. Since 1 centimeter equals 0.01 meters, converting 2.3 cm/s results in 0.023 m/s. Understanding unit conversion is vital because using the incorrect units can lead to calculation mistakes. Always ensure units are uniform across the calculations to achieve reliable results.

Force and Velocity

Force and velocity are fundamental concepts connecting to motion in physics. Force, measured in newtons (\(N\)), is what causes an object to change its state of motion or direction. For a fly climbing against gravity, the force is the gravitational pull opposed by the fly's legs gripping the pane.

Velocity, measured in meters per second \((m/s)\), defines the speed and direction of an object's movement. It’s crucial to differentiate it from speed, as velocity includes direction, making it a vector quantity.
When calculating power output, the key formula that connects these concepts is \( P = Fv \), where force and velocity act in the same direction. The product of force (gravitational in this context) and the fly's velocity gives the power output, which measures how effectively it uses energy to climb vertically. Recognizing the relationship between these concepts aids in understanding how power is distributed in physical systems.