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.

 

[HalfLostAndSearching]

I would approach this problem by using the principles of conservation of energy and work. In this case, the gravitational potential energy of the slender rod can be calculated by considering the work done by gravity as the rod moves from its initial position at theta = 0 to its final position at theta = theta_0.

To do this, I would divide the rod into small segments and calculate the work done by gravity on each segment as it moves. The work done by gravity on a segment can be calculated as the product of its mass, the gravitational acceleration, and the vertical displacement of its center of mass. This can be expressed as W = mgh, where m is the mass of the segment, g is the gravitational acceleration, and h is the vertical displacement of the segment's center of mass.

By summing up the work done by gravity on each segment, we can calculate the total change in gravitational potential energy of the rod. This approach is analytical and does not rely on intuition, making it a reliable method for calculating the change in gravitational potential energy for any given position of the rod.

In the case of the specific scenario given, where the rod starts at rest and the datum is chosen at theta = 0, the calculation simplifies to mg(0.6+0.2)sin(theta_0)/2, as stated in the attempt at a solution. However, this approach can be used for any general case, as long as the work done by gravity on each segment is properly calculated.