Change in gravitational potential energy for a slender

[zeralda21]

Homework Statement

I'll provide a picture for a clearer view: http://i.imgur.com/wkXPcJn.jpg

Suppose that the slender rod starts at rest at theta = 0. For convenience we chose the datum at theta = 0.
Now I want to calculate the gravitational potential energy at a later instant when theta = theta_0. But it's tricky since some points of the rod have moved a distance (0.6+0.2)sin(theta_0) and some points have not moved at all. So how does one deal with this case? I have a solution but how should I do in the general case?

Just to be clear: I am looking for an analytical approach to it that does not involve some intuition because that can
be dangerous...

The Attempt at a Solution

Think of the slender rod as a huge amounts of points uniformly spread. For each point at one side of the mass center there is a point on the other side of the mass center so that the distance between these are the distance from O to the mass center. Hence it should be mg(0.6+0.2)sin(theta_0)/2

 

[SammyS]

Homework Statement

I'll provide a picture for a clearer view: http://i.imgur.com/wkXPcJn.jpg

Suppose that the slender starts at rest at theta = 0. For convenience we chose the datum at theta = 0.
Now I want to calculate the gravitational potential energy at a later instant when theta = theta_0. But it's tricky since some points of the slender have moved a distance (0.6+0.2)sin(theta_0) and some points have not moved at all. So how does one deal with this case? I have a solution but how should I do in the general case?

Just to be clear: I am looking for an analytical approach to it that does not involve some intuition because that can
be dangerous...

The Attempt at a Solution

Think of the slender as a huge amounts of points uniformly spread. For each point at one side of the mass center there is a point on the other side of the mass center so that the distance between these are the distance from O to the mass center. Hence it should be mg(0.6+0.2)sin(theta_0)/2

A slender what ?

The word slender is not a noun.

 

[zeralda21]

I am sorry. I thought slender and rod is the same thing. I mean a slender rod

 

[SammyS]

Yes.

mg((0.6+0.2)/2)sin(θ₀) is a valid expression for the gravitational potential energy of the rod.