Temperature difference physics Problem

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Homework Help Overview

The problem involves the temperature difference between a body and a constant medium, exploring the rate of change of this difference over time. It is framed within the context of Newton's Law of Cooling, specifically examining how the temperature difference decreases from 40 °C to 20 °C in 5 minutes, and seeks to determine the value of a constant k and the time required for the temperature difference to reach 10 °C.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • Participants discuss the nature of the constant k, questioning whether it should be considered negative or positive based on the behavior of the temperature difference over time. There are attempts to clarify the implications of the temperature difference decreasing and its relation to the equation provided.

Discussion Status

The discussion is active, with participants providing insights into the interpretation of k and its sign. Some participants suggest that the temperature difference's decrease indicates a positive k, while others explore the implications of the initial conditions. There is a recognition that understanding the relationship between the temperature difference and time is crucial for solving the problem.

Contextual Notes

Participants note that the problem is framed in a way that requires addressing both parts (a) and (b) sequentially, although some suggest that part (b) could be approached independently with the right understanding of the exponential decay involved.

cscott
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The temperature of a body differs from that of a medium, whose temperature is kept constant, by 40 °C. In 5 min, this difference is 20 °C. (a) What is the value of k? (b) In how many minutes will the difference in temperature be 10 °C?

[tex]T_B-T_M = \left ( T_0-T_M \right )e^{-kt}[/tex]

Do we know k is negative only because of what's said in part (b)?
 
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I would say that the value of k is negative because it is the difference in temperature between the object and environment that is decreasing with time. It wouldn't make any difference whether the object was initially hotter or cooler than its surrrounding environment.
 
Hammie said:
I would say that the value of k is negative because it is the difference in temperature between the object and environment that is decreasing with time. It wouldn't make any difference whether the object was initially hotter or cooler than its surrrounding environment.

Yeah, I should have quoted the part dealing with 5 minutes later. Thanks.
 
cscott said:
The temperature of a body differs from that of a medium, whose temperature is kept constant, by 40 °C. In 5 min, this difference is 20 °C. (a) What is the value of k? (b) In how many minutes will the difference in temperature be 10 °C?

[tex]T_B-T_M = \left ( T_0-T_M \right )e^{-kt}[/tex]

Do we know k is negative only because of what's said in part (b)?

Actually we know that k is positive (so that -k is negative) because of the fact that the difference between the two temperatures is decreasing (from 40 to 20 in 5 minutes).

After 5 minutes, TB- TM= 20 and T0- TM= 40 so the equation, with t= 5 says
20= 40e-5k. It should be easy to solve that for k and then solve 10= 40e-kt, with that value of k, for t.

By the way, you don't really need to answer (a) in order to answer (b) (although you do have to answer (a) anyway!). If the temperature difference drops from 40 to 20 in 5 minutes, then it halves in in 5 minutes. In another 5 minutes, it will halve again! (That's a basic property of exponentials).
 

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