Calculation of the rise in the temperature of a truck brake

[KedarMhaswade]

Homework Statement

Calculate the temperature increase of 100 kg of brake material with an average specific heat of 800 J/kg⋅ºC if the material retains 10% of the energy from a 10,000-kg truck descending 75.0 m (in vertical displacement) at a constant speed. (see: https://openstax.org/books/college-physics-ap-courses/pages/14-2-temperature-change-and-heat-capacity)

Relevant Equations

Equating the loss of potential energy of the truck to the gain of the internal energy of the brake material seems alright, but is that what is required? What about friction and loss of heat to atmosphere during the descent?

The text gives the answer as 92ºC. The answer is arrived at by doing $Q=Mgh=mc\Delta T$. But it is unclear to me if they are the same. I checked the coefficient of friction and it definitely seems to be considerable. Is the entire PE lost by the truck going to result in increasing the temperature of the brake material, or is it only 10% of the former (resulting in an answer of 9.2ºC)?

 

[haruspex]

What about friction and loss of heat to atmosphere during the descent?

I checked the coefficient of friction and it definitely seems to be considerable.

What friction do you have in mind, other than that which heats the brakes?

or is it only 10%

It does seem that the 92C answer overlooks the advice that only 10% of the generated heat is retained.

 

[KedarMhaswade]

What friction do you have in mind, other than that which heats the brakes?

Should we not take into account the rolling friction of the tire on the surface? Agreed, it is negligible perhaps, but it is not zero. Rudimentary calculations show that at a descent of $75\space m$ under gravity on a frictionless surface, a body will be moving at an instantaneous speed of about $40\space m/sec$. Will the body be moving with the same speed in the presence of rolling friction as well?

It does seem that the 92C answer overlooks the advice that only 10% of the generated heat is retained.

I thought so too. It does look like 90% of the energy is lost to atmosphere. The funny part is that the book goes on to acknowledge such a high temperature rise by saying "This temperature is close to the boiling point of water. If the truck had been traveling for some time, then ...". This tells me that perhaps the rise is indeed close to the boiling point of water at 1 atm.

Should I submit an errata to OpenStax, or am I to ignore the 10% heat retention information in the problem statement?

 

[haruspex]

Should we not take into account the rolling friction of the tire on the surface?

Ok, that's what I prefer to call rolling resistance. With the brakes off, how much of a slope would lead a truck to start rolling? Not much, I think.

Should I submit an errata to OpenStax

Might as well.