Finding the angular velocity of a rotating rod

[LuigiAM]

Hi everyone, sorry for bothering y'all again but here is another problem that I'm struggling with!

This is the question and the professor's solution guide:

When I solve the problem, I always end up getting 7.67 rad/s instead of 3.83 rad/s.

My understanding of the solution is this:

This is a conservation of energy problem.

When the rod is suspended at 60 degrees, it has potential energy (U = m g h). When it is released from its position, the potential energy turns into kinetic energy. The potential energy has completely turned into kinetic energy when the rod is horizontal. So, at the horizontal, the potential energy has turned into rotational kinetic energy E_k = I w².

So I have:

m g h = 1/2 I w²

The moment of inertia for a rod is 1/12 m L². Plugging it in the equation:

m g h = 1/2 (1/12 m L²) w²

I can cancel out m from both side and remove L² since L = 1. The equation becomes:

g h = 1/24 w²

w² = 24(9.8)(h)

h is the height at which the rod is raised. Since half of the rod is 0.5 meters and it is at a 30 degree angle to the horizontal, the height h will be 0.5 times the sine of 30 degrees, so h will be 0.25 m.

Plugging that in for h:

w² = 24(9.8)(0.25)
w² = 58.8
w = 7.67 rad/s

I know I'm doing something wrong but I'm not sure where I'm going off track. I suppose it's probably when I'm solving for h?

I have a lot of difficulty understanding the professor's solution, but at the end he has the equation:

w² = 9.8(0.5)(3)

I understand he gets a different number, but I don't see where that comes from? I read the solution 20 times but I don't understand his process. I remember that in class we did a rotating rod problem and the idea was just a conservation of energy from potential energy to kinetic energy.

I also don't see where he calculates the value for h in the solution? I looks like something that should be an easy problem but I feel lost here :(

 

[Doc Al]

The potential energy has completely turned into kinetic energy when the rod is horizontal. So, at the horizontal, the potential energy has turned into rotational kinetic energy E_k = I w².

The rod is both rotating and translating. Don't forget the translational KE of the center of mass.

 

[LuigiAM]

The rod is both rotating and translating. Don't forget the translational KE of the center of mass.

Hi, thanks for your response.

I don't understand how the rod is translating? I mean, both ends move, but it is rotating at the center of mass so the center of mass itself is stationary? At least that's my understanding? The problem says that the rod is uniform so I didn't think we needed to consider the translation of the moving parts

 

[LuigiAM]

Oh... oh my... I think I get it. The rod is actually falling down to the floor at the same time it is rotating. Wow, I feel dumb now. I can't believe it took me so long to see it.

Ok, back to the drawing board for me! Sorry about this

 

[Doc Al]

The rod is actually falling down to the floor at the same time it is rotating.

Exactly!

 

[LuigiAM]

Phew, ok, here is my second attempt.

I get to the good answer this time at 3.83 rad/s.

However, it feels like I used a different procedure than the solution? I'm not sure I understand what is happening in the solution. The solution talks about work done but I don't understand how work enters into the equation? Is it correct to use conservation of energy?

 

[Doc Al]

I get to the good answer this time at 3.83 rad/s.

Good!

However, it feels like I used a different procedure than the solution? I'm not sure I understand what is happening in the solution. The solution talks about work done but I don't understand how work enters into the equation? Is it correct to use conservation of energy?

The work done by gravity and the changes in gravitational PE are just two ways of describing the same thing. So it is your choice. (The change in gravitational PE equals the negative of the work done by gravity.)

 

[LuigiAM]

Yeah thanks! I tend to get panic attacks when the solutions get to the answer in a different way because it makes me think I did something wrong but I guess its not always the case