Problem 2 - Pearson Physics

Question

A swimmer does \(4.3 \times 10^{5} \mathrm{~J}\) of work and gives off \(1.7 \times 10^{5} \mathrm{~J}\) of heat during a workout. Determine \(\Delta E, W\), and \(Q\) for the swimmer.

Step-by-Step Solution

Answer

\(\Delta E = 2.6 \times 10^{5} \text{ J}\), \(W = 4.3 \times 10^{5} \text{ J}\), \(Q = -1.7 \times 10^{5} \text{ J}\)

Step 1: Identify Given Values

We are given that the swimmer does \(4.3 \times 10^{5} \, \text{J}\) of work and gives off \(1.7 \times 10^{5} \, \text{J}\) in the form of heat. These correspond to work \(W = 4.3 \times 10^{5} \, \text{J}\) and heat \(Q = -1.7 \times 10^{5} \, \text{J}\) (negative because it is energy released).

Step 2: State the First Law of Thermodynamics

The first law of thermodynamics is given by the formula \(\Delta E = Q + W\), where \(\Delta E\) is the change in internal energy, \(Q\) is the heat exchanged, and \(W\) is the work done.

Step 3: Substitute Known Values into the Formula

Substitute the given values into the first law of thermodynamics equation: \(\Delta E = (-1.7 \times 10^{5} \text{ J}) + (4.3 \times 10^{5} \text{ J})\).

Step 4: Perform the Calculation

Calculate \(\Delta E\) by solving \((-1.7 \times 10^{5}) + (4.3 \times 10^{5})\). This results in \((4.3 - 1.7) \times 10^{5}\), which simplifies to \(2.6 \times 10^{5} \, \text{J}\).

Step 5: Summarize the Solution

The change in internal energy \(\Delta E\) is \(2.6 \times 10^{5} \, \text{J}\), the work done \(W\) is \(4.3 \times 10^{5} \, \text{J}\), and the heat exchanged \(Q\) is \(-1.7 \times 10^{5} \, \text{J}\).

Key Concepts

Work-Energy Principle

The work-energy principle is a fundamental concept in physics which states that work done by all forces acting on a system results in a change in that system's energy. Simply put, when energy is applied to an object through work, the object's energy changes.
This principle plays a crucial role in understanding how the swimmer in our exercise exerts a force to move through water. The swimmer does a total of \(4.3 \times 10^{5} \, \text{J}\) of work, which means energy has been transferred from the swimmer's muscles to the water. This energy aids the movement, propelling the swimmer forward.

• Work done by the swimmer equates to energy applied to their body and the water.
• Understanding work helps us calculate energy changes in various situations.

By comprehending the work-energy principle, we can better understand how energy is used and transformed during physical activities like swimming.

Heat Transfer

Heat transfer refers to the movement of thermal energy from one object or system to another due to a difference in temperature. In relation to the first law of thermodynamics, heat transfer affects the internal energy of a system.
In the swimmer's case, they release \(1.7 \times 10^{5} \, \text{J}\) of heat as their body adjusts to maintain temperature. This release acts as a form of energy exchange, helping keep the swimmer's body cool during exertion.

• A negative sign indicates that the swimmer is giving off heat.
• Heat transfer can be conductive, convective, or radiative.

Recognizing how heat transfers during exercise helps us understand the body's response during physical activity and how it affects overall energy consumption.

Internal Energy Change

Internal energy change, denoted as \(\Delta E\), is a measure of how the total energy within a system varies, mainly caused by heat exchange and work done. According to the first law of thermodynamics, this change in internal energy is expressed through the equation \(\Delta E = Q + W\).
For the swimmer, \(\Delta E\) is calculated by combining the work done and the heat given off:

\[\Delta E = (-1.7 \times 10^{5} \, \text{J}) + (4.3 \times 10^{5} \, \text{J}) = 2.6 \times 10^{5} \, \text{J}\]

• The positive \(\Delta E\) implies an increase in the swimmer's total internal energy.
• This energy change is the result of both mechanical work and thermal processes.

Understanding internal energy change provides insights into how energy is conserved and transformed within a system under various conditions.