Work-Kinetic and Work-Potential theorems relation

[aljan9559]

TL;DR

My question is about the relation between the Work-Kinetic (W = change in PE) and Work-Potential theorem (W = change in KE). Does the change in KE and change in PE in the same scenario result in the same work value?

In a scenario of a free-falling object in a vacuum on earth, the object will be acceleration towards the earth. According to the theorem of Work-Kinetic and Work-Potential:

* Since the object is accelerating towards the earth, we know that the object's Kinetic energy is increasing because the velocity is increasing. According to the Work-Kinetic theorem, there is positive work done on the object.
W = KE2 - KE1, and KE2 > KE1 therefore, W is positive.

* Now, my question is if we were to apply the same concept on the Work-Potential theorem with the exact same scenario, would the work be positive as well? When thinking about it, the free-falling object in a vacuum, by time, is getting closer to Earth and therefore the height is decreasing. So, W = PE2 - PE1, and PE2 < PE1 and that would result in work (W) to be (-) negative. Why is that? Isn't supposed to be that the work is positive in that scenario, why are the results different when using the Work-kinetic theorem as opposed to using the Work-Potential theorem to find the work?

Thanks, I appreciate your time.

 

[PeroK]

Let me start with an analogy. You have $100 in one bank account and $500 in a savings account. You withdraw $50 from your savings and deposit it your other bank account. Was that $50 a positive or negative transaction? In other words, did that transaction involve plus or minus $50?

In your scenario it's clear that the object increases its KE and loses gravitational PE. It's no more complicated than that.

The gravitational field/force, therefore, does positive work on the object.

That said, I must confess I've never heard of a work-potential theorem.

 

[kuruman]

In your equation W = PE2 - PE1, if the symbol W stands for the work done by gravity on the falling object, you have written it incorrectly. It should be written as W = - (PE2 - PE1). The change in potential energy is the negative of the work done by the conservative force. That comes from the definition of potential energy.

 

[aljan9559]

I found my textbook mentioning that about Work-kinetic:

So, do they mean that we shouldn't be applying W = PE2-PE1 and W = KE2 - KE1 in one scenario?

Also, this is how the book built up the W = PE2 - PE1 equation.

 

[kuruman]

In this particular example there are two forces doing work on the book, the person lifting the book and gravity. Suppose the book is lifted at constant speed so that its kinetic energy does not change. Then
1. The work done by the person on the book is W_person = + mgh
2. The work done by gravity on the book is W_grav. = - mgh
3. The change in potential energy is the negative of the work done by gravity, PE2 - PE1 = -(-mgh) = + mgh.

In this particular example it turns out that the change in potential energy is also the work done by the person but that cannot be generalized into a rule. The rule is the change in potential energy is equal to the negative of the work done by conservative forces only. The force exerted by the person is not conservative but gravity is.