# Uncertainty in Newton's law of cooling

• sunmoonlight
In summary, the conversation is discussing the uncertainty in Newton's law of cooling, specifically in the value of k. The relevant equations are also mentioned. The uncertainty is calculated using a root-sum-square approach and the question of finding uncertainty involving logs is raised.
sunmoonlight
Homework Statement
Uncertainty in Newton's law of cooling
Relevant Equations
T(t) = = 𝑇_𝐴+(𝑇_𝑜−𝑇_𝐴)𝑒^(−𝑘𝑡)
I'm finding the uncertainty of k, given that each temperature has an uncertainty of +/- 0.5 degress.

sunmoonlight said:
Homework Statement: Uncertainty in Newton's law of cooling
Relevant Equations: T(t) = = 𝑇_𝐴+(𝑇_𝑜−𝑇_𝐴)𝑒^(−𝑘𝑡)

I'm finding the uncertainty of k, given that each temperature has an uncertainty of +/- 0.5 degress.
You will also need approximate values for the temperatures.
Per forum rules, please show some attempt.

say the T(O) = 90 +/- 0.5, T(t): 60 +/- 0.5, TA = 10 +/- 0.5, temp difference (T(t) - TA) is 50 degrees +/- 0.5, t= 100s
1. Is the uncertainty for ln (T(t) - TA) = 1/2*(ln50.5 - ln49.5) = +/-0.01?
2. If you substitute the values into the eqt, you get k = (ln50/80)/-100, so what's the uncertainty for k (like how do you find uncertainty involving logs?)

sunmoonlight said:
say the T(O) = 90 +/- 0.5, T(t): 60 +/- 0.5, TA = 10 +/- 0.5, temp difference (T(t) - TA) is 50 degrees +/- 0.5, t= 100s
1. Is the uncertainty for ln (T(t) - TA) = 1/2*(ln50.5 - ln49.5) = +/-0.01?
2. If you substitute the values into the eqt, you get k = (ln50/80)/-100, so what's the uncertainty for k (like how do you find uncertainty involving logs?)
There are different concepts of uncertainty. An engineer worried about engineering tolerances would just look at the combinations of the extreme values. A scientist would take the given uncertainties as standard deviations in normal distributions and use root-sum-square approaches to combine them. I assume you are looking for the latter.

Can you find the uncertainty in ##e^{-kt}##?

## What is Newton's law of cooling?

Newton's law of cooling states that the rate of change of the temperature of an object is proportional to the difference between its own temperature and the ambient temperature, provided the difference is small. Mathematically, it can be expressed as: dT/dt = -k(T - T_env), where T is the temperature of the object, T_env is the ambient temperature, and k is a constant.

## What are the common sources of uncertainty in Newton's law of cooling?

Common sources of uncertainty include variations in the ambient temperature, inaccuracies in the initial temperature measurement, heat loss to the surroundings not accounted for by the simple model, and the assumption that the cooling constant k remains constant over time.

## How can measurement errors affect the application of Newton's law of cooling?

Measurement errors can lead to inaccuracies in determining the initial temperature, the ambient temperature, and the temperature at various time points. These inaccuracies can result in incorrect estimations of the cooling constant k and consequently affect the prediction of the cooling process.

## How is the cooling constant (k) determined, and what are its uncertainties?

The cooling constant k is typically determined experimentally by measuring the temperature of an object over time and fitting the data to the Newton's law of cooling equation. Uncertainties in k arise from measurement errors, environmental fluctuations, and assumptions made during the fitting process, such as neglecting heat loss mechanisms other than convection.

## Can Newton's law of cooling be applied to all cooling scenarios?

No, Newton's law of cooling is most accurate for small temperature differences and when the primary mode of heat transfer is convection. For large temperature differences, or when other modes of heat transfer (such as radiation or conduction) are significant, the law may not provide accurate predictions, and more complex models may be required.

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