1. What is Newton's Law of Cooling?

Newton's Law of Cooling describes the rate at which an object's temperature changes when it is in contact with a medium at a different temperature. It states that the rate of heat loss is proportional to the temperature difference between the object and its surroundings.

2. How Does the Calculator Work?

The calculator uses Newton's Law of Cooling differential equation:

Where:

• \( \frac{dT}{dt} \) — Rate of temperature change (°C/s)
• \( k \) — Cooling constant (1/s)
• \( T \) — Current temperature of the object (°C)
• \( T_a \) — Ambient temperature (°C)

Explanation: The negative sign indicates that the object cools when T > T_a and warms when T < T_a. The cooling constant k depends on the object's properties and the surrounding medium.

3. Importance of Newton's Law of Cooling

Details: This law is fundamental in thermodynamics and has applications in engineering, meteorology, food science, and forensic science for estimating time of death.

4. Using the Calculator

Tips: Enter the cooling constant k in 1/s, current temperature T in °C, and ambient temperature T_a in °C. The cooling constant must be positive.

5. Frequently Asked Questions (FAQ)

Q1: What factors affect the cooling constant k?
A: k depends on the object's surface area, material properties, and the heat transfer coefficient of the surrounding medium.

Q2: Is Newton's Law of Cooling accurate for all situations?
A: It's most accurate for small temperature differences and when heat transfer occurs primarily through convection.

Q3: How is this related to BC Calculus?
A: This is a classic application of first-order linear differential equations studied in AP Calculus BC.

Q4: Can this be used for heating as well as cooling?
A: Yes, the equation works for both heating and cooling depending on the temperature difference.

Q5: What are typical values for the cooling constant k?
A: k values vary widely but are typically between 0.001 and 0.1 1/s for common objects in air.