# Why and How is there heat loss due to friction?

In this question, the answer is given 432 which is double of the expected 216.

This appeared counterintuitive.

Why is the work done by the motor double of the difference in final and initial kinetic energies?
Friction is an internal force therefore it should not have any effect on the work done.

The teacher says it is due to heat loss. This leads to a more general question.

Why and How (molecular mechanisms)does friction cause loss of energy as heat? Why is it equal to gain in energy of crate?
There should not be be any loss in energy as heat if energy lost by belt is gained by block and that is provided by motor. This should also hold for other things like walking or driving

Assuming there is loss of heat energy, is it just friction or do other forces like tension give the same result under similar conditions?

Edit:
This Sand on conveyer belt question does not answer the how part of my question. It does not mention what molecular interactions cause the loss of energy. (Or why if two forces are applied in opposite directions on two bodies the work done is double even if one is negative and other positive.)
It also doesn't specify whether or not friction causes a transfer of energy from belt to crate.

It does not talk why exactly double of the energy gained by the crate is the required work done

I also asked if all forces behave this way or if it was just friction which remains unanswered

• Commented May 15, 2023 at 21:06
• Please review the earlier discussion closely, including the clarifying exchanges in the comments. You may also wish to review the many discussions I linked. You can posit any dissipative mechanism you like to dampen the oscillation when the crate lands—friction, mechanical hysteresis, etc. (There must be oscillation because objects can't accelerate infinitely fast.) But something must apply damping. Look at it from the perspective of the belt: You have a crate that starts off with relative speed that is completely converted to thermal energy by the time the crate is moving with the belt. Commented May 16, 2023 at 0:35
• Read the answer by Farcher, especially the part that says: "During the acceleration phase when there is slippage between the belt and the sand heat/thermal energy is generated and the rate of heat generation is the difference between your two answers." Commented Jun 11, 2023 at 21:02

This is easier to understand by looking at the mechanical power entering the system $$P=\vec F \cdot \vec v$$ where $$\vec v$$ is the velocity of the material of the system at the point of application of the force $$\vec F$$.

Let $$\hat i$$ be a unit vector pointing to the left. We will analyze only during the time that the crate is accelerating, $$0.

For the belt the velocity is $$v_{b} \ \hat i$$ and the kinetic friction force acting on the belt is $$-F_{k} \ \hat i$$. So the mechanical power is $$P_b=-v_b \ F_k$$. Since this is negative it means that mechanical power is leaving the belt at a rate of $$P_b$$.

For the crate the velocity is $$v_c \ \hat i$$ where $$v_c = a t = t \ F_k/m = t \ \mu_k \ g$$. On the crate the force is $$F_k \ \hat i$$ so the mechanical power is $$P_c=v_c \ F_k = F_k \ a \ t$$. Note that this is positive, meaning that mechanical power is entering the crate at a rate of $$P_c$$, and note further that that rate is not constant but it is increasing linearly over time.

So there is mechanical power leaving the belt at a constant rate and mechanical power entering the crate at a linearly increasing rate. Thus the total mechanical power is negative and goes to zero.

The negative total mechanical power is mechanical power that leaves the belt but does not enter the crate. That power goes to heat.

Why is the work done by the motor double of the difference in final and initial kinetic energies?

The work is the integral of the power, or the area under the curve in the plot. The difference in KE is the area of the triangle and the work done is the area of the rectangle. The triangle is half the area of the rectangle.

Friction is an internal force therefore it should not have any effect on the work done.

Whether it is internal or external is not relevant. The fact is that the total mechanical power is negative, so mechanical power is being converted to something else (heat).

Why and How (molecular mechanisms)does friction cause loss of energy as heat?

Heat isn’t defined at the molecular level. You need to have a very large number of molecules to even discuss heat.

is it just friction or do other forces like tension give the same result under similar conditions?

The same thing happens in many other contexts. Whenever you have one constant power and one linearly increasing power then you will have a factor of 2 conversion of energy from one form to another. Links have been provided in the comments.

what molecular interactions cause the loss of energy

The “loss” of energy is due to the fact that the molecules of the belt are moving at a different velocity than the molecules of the crate. So the equal and opposite forces produce different power.

specify whether or not friction causes a transfer of energy from belt to crate

Yes, friction causes a transfer of energy from belt to crate. The amount of mechanical power leaving the belt is greater than the amount of mechanical power entering the crate.

why exactly double of the energy gained by the crate is the required work done

The area of the rectangle is exactly double the area of the triangle.

if all forces behave this way or if it was just friction which remains unanswered

Friction is not unique in this regard. A common other one is charging a capacitor.