Average tension force acting on a simple pendulum

[haruspex]

in what way is it wrong?

Consider e.g. average fuel consumption. That should be an average wrt distance. If you were to average over time instead, consider a journey at varying speed. While going slowly, more time is spent over each equal distance, so that would make those fuel measurements count more.

 

[Pushoam]

Consider e.g. average fuel consumption. That should be an average wrt distance. If you were to average over time instead, consider a journey at varying speed. While going slowly, more time is spent over each equal distance, so that would make those fuel measurements count more.

Yes, this means that if I go slowly, I will spend more fuel, isn't it so?
Does this mean that the spent fuel is independent of speed?
If I consider conservation of energy, the more speed means more kinetic energy, so more fuel is needed as fuel is the energy which gets converted into the kinetic energy for a car to reach greater speed. So, this means that the rate of fuel spent per unit time that is input power is more if the car is accelerating.

I think you suggested to consider an unaccelerated moving car.

If a car is already moving at constant speed v, then the fuel is needed to counter the friction force, not to keep the car moving at speed v. And the friction force is independent of speed. The fuel is equal to the |friction force| *distance, which is independent of v.

So, the amount of consumed fuel depends upon the traveled distance, not time or speed.
So, the average of consumed fuel should be calculated wrt to distance.

Now, I have to answer, why should $<\theta^2>$ be calculated wrt time not $\theta$?

 

[Pushoam]

Now, I have to answer, why should <θ²> be calculated wrt time not θ?

The first thought which I am getting is : θ² depends on θ and θ depends on time t explicitly. So, <θ²> should remain same. But, I know that it is not so as I found them different after calculation. So, I leave this expectation.
Now, why should <θ²> be calculated wrt time not θ?
I am not getting how the fuel example helps to know the answer to the above question.
Is this because while calculating <T>, I have calculated average of other terms wrt time t, not θ?

Another thought:
The question asks to calculate <T> wrt time t over one time - period, not θ. So, <θ²> should be calculated wrt time t,not θ.