# Homework Help: Falling Plate

1. Aug 10, 2013

### postfan

1. The problem statement, all variables and given/known data

A uniform square plate ABCD has mass 0.8 kg and side length .8 m. The square is pivoted at vertex D and initially held at rest so that sides AB and CD are horizontal (see the diagram). After it is released, the plate swings downward, rotating about the pivot point. Resistive force can be neglected. The acceleration due to gravity is g = 9.8 m/s2. The rotational inertia of a square plate of side d relative to the axis perpendicular to the plate and passing through the center of mass is md^2/6.

Find the rotational inertia of the plate relative to the axis of rotation.
Find the angular speed of the plate at the moment when BD is horizontal.
Find the linear speed of B at the moment when BD is horizontal.
Find the linear speed of B at the moment when B is at the bottom position.

2. Relevant equations

3. The attempt at a solution
I used the parallel axis theorem to find the moment of inertia around the axis of rotation. I got
1/6(.8)(1.2)^2+.8(sqrt(2)/2)^2=.592 but that isn't correct.

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Last edited: Aug 10, 2013
2. Aug 10, 2013

### TSny

What does the quantity √(2)/2 represent?

3. Aug 10, 2013

### barryj

never mind

Last edited: Aug 10, 2013
4. Aug 10, 2013

### postfan

√(2)/2 represents the distance from point D to the center of mass.

5. Aug 10, 2013

### TSny

Shouldn't the 1.2 m come into play here?

6. Aug 10, 2013

### postfan

Oh ok, that makes sense. I got .768, is that right?

7. Aug 10, 2013

### TSny

I believe that's correct. (Add units of course.)

8. Aug 10, 2013

### postfan

Ok, then how do I start solving part B?

9. Aug 10, 2013

### TSny

Is anything conserved while the plate swings down?

10. Aug 10, 2013

### postfan

Energy,right?

11. Aug 10, 2013

### TSny

Right. Give it a shot.

12. Aug 10, 2013

### postfan

Ok I got 4.937. Is that right?

13. Aug 10, 2013

### TSny

I don't know, I haven't worked it out. Please show your work and we can see if you are setting it up correctly.

14. Aug 10, 2013

### postfan

m*g*h=.5*I*w^2

m*g*h=.5*.768*m*r^2*(v/r)^2

g*h=.384*v^2

9.8*.6=.384*v^2

v=3.91 (not 4.937 due to arithmetic error)

15. Aug 10, 2013

### TSny

But note that in the second question, you are asked to find the angular speed rather than the linear speed.

Also, I don't agree with the value that you used for h in the calculation. How far does the center of mass of the square move downward for this question?

16. Aug 10, 2013

### postfan

I still think that its .6. Can you tell me why I'm wrong?

17. Aug 10, 2013

### TSny

I'm a little confused about the value of the side length of the square plate. I thought it was 1.2 m, but now when I look at your original post, it appears to be 0.8 m.

But either way, h would not be 0.6 m for the second question. Did you draw a picture showing the orientation of the square at the moment of release and then another picture for when BD is horizontal? Be sure to mark the center of mass in each case.

Since a diagram wasn't included, I am interpreting the geometry as best I can from the wording of the problem.

Last edited: Aug 10, 2013
18. Aug 10, 2013

### TSny

OK, now I see the picture. I had misinterpreted it. I thought that initially point B was directly above D so that BD would be a side of the square. But, BD is a diagonal of the square. So, if the length of a side is 1.2 m, then I agree that h would be 0.6 m for the second question. Sorry about that.

19. Aug 10, 2013

### postfan

For parts 3 and 4 is h= to .8 and .8(2+sqrt(2)/2) respectively?

20. Aug 10, 2013

### TSny

h for part 3 should be the same as for part 2.

Can we clarify if the length of the side of the square is 1.2 m or is it 0.8 m?

21. Aug 10, 2013

### postfan

Length is .8.

22. Aug 10, 2013

### TSny

Oh Oh. That's going to change the value of I that you calculated using 1.2 m:

23. Aug 10, 2013

### postfan

I know, I already accounted for it.

24. Aug 10, 2013

### postfan

So does h= .8(2+sqrt(2)/2) in part 4?

25. Aug 10, 2013

### TSny

OK. I agree with your value of h for part 3, but not for part 4. Can you explain how you got your expression for part 4?