Calculating Rotational Inertia and Speed of a Square Plate

[postfan]

Homework Statement

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.

Homework Equations

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.

 

[TSny]

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.

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

 

[barryj]

never mind

 

[postfan]

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

 

[TSny]

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

Shouldn't the 1.2 m come into play here?

 

[postfan]

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

 

[TSny]

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

 

[postfan]

Ok, then how do I start solving part B?

 

[TSny]

Is anything conserved while the plate swings down?

 

[postfan]

Energy,right?

 

[TSny]

Right. Give it a shot.

 

[postfan]

Ok I got 4.937. Is that right?

 

[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.

 

[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)

 

[TSny]

Your set up looks good!

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?

 

[postfan]

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

 

[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.

 

[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.

 

[postfan]

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

 

[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?

 

[postfan]

Length is .8.

 

[TSny]

Length is .8.

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

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.

 

[postfan]

I know, I already accounted for it.

 

[postfan]

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

 

[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?

 

[TSny]

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

I don't agree with the 2 that is shown in red above.

 

[postfan]

Is it .8(1+sqrt(2)/2)?

 

[TSny]

OK, I must be getting tired. The value of h for parts 2 and 3 is not the length of the side of the square. So, h is not .8 for parts 2 and 3. It's a certain fraction of the length of the side.

For part 4, you still don't quite have the right answer.

 

[postfan]

I relooked at it and still got .8(1+sqrt(2)/2). Can you tell me why I am wrong?

 

[TSny]

I relooked at it and still got .8(1+sqrt(2)/2). Can you tell me why I am wrong?

I don't know why you are wrong because I don't know the thought process that you used to get your answer. That's why I wanted you to explain how you got your answer. Your answer is pretty close to the correct answer. It might help if you answer the following:

(A) How far does the center of mass move downward in going from question 1 to question 2?

(B) How much additional distance does it go down in going from question 2 to question 4?

It looks to me that you probably have the correct answer to (B) but not the correct answer to (A).

 

[postfan]

From 1 to 2 I found that the center of mass move downs by .4.
From 2 to 3 I found that the center of mass move down by .4+ sqrt(2)/2*.8.

 

[TSny]

From 1 to 2 I found that the center of mass move downs by .4.
From 2 to 3 I found that the center of mass move down by .4+ sqrt(2)/2*.8.

I agree that h₁ = .4 for the first part. I don't agree that h₂ = .4+ sqrt(2)/2*.8 for the additional distance from question 2 to question 4.

 

[postfan]

Does h2 = .8+ sqrt(2)/2*.8

 

[TSny]

Does h2 = .8+ sqrt(2)/2*.8

No. See the attached diagram. As corner B moves from B' to B'', the center of mass moves downward a distance h₂. You can see that h₂ is half the diagonal of the square.

 

[postfan]

Ok so for part 4 h= sqrt(2)/2*.8, right?

 

[TSny]

Ok so for part 4 h= sqrt(2)/2*.8, right?

That represents the correct value of h₂. What is the total h in going from where the plate is released from rest to where B is at the bottom?

 

[postfan]

The total h is equal to .4+sqrt(2)/2*.8.

 

[TSny]

Yes, that's right.

 

[postfan]

Ok then we use the equation mgh=.5mv^2+.5Iw*2, substitute values convert angular velocity to linear velocity and then solve for v, right?

 

[TSny]

Yes. [EDIT: NO]

 

[TSny]

Sorry, it's late here and I'm half asleep! The total kinetic energy is just (1/2)Iw² (as you had in post #14). So,
mgh = (1/2)Iw².

 

[postfan]

Why is that so? The question is asking for the linear speed of B.

 

[TSny]

The plate is in pure rotation about D. So, as long as you use the moment of inertia about D, the kinetic energy can be expressed as just rotational KE about D.

Equivalently, you could express the kinetic energy as kinetic energy due to translation of the center of mass plus kinetic energy of rotation about the center of mass:

KE = (1/2)mV_cm² + (1/2)I_cmω². Here you would use the moment of inertia about the center of mass rather than about point D.

Either way will yield the same answer.

Once you find ω you can then find the linear speed of B.

 

[postfan]

I got the answer, but I am still confused about when/how I can just use (1/2)Iw2 instead of the full (1/2)mV^2 + (1/2)Iω^2. I know it is rather late, so would you rather want to talk about it tomorrow?

 

[TSny]

As I said before, you can use either KE = (1/2)I_Dω² or KE = (1/2)mV_cm² + (1/2)I_cmω². Both are correct.

In this problem, the first question was to find I_D. So, it will be easier to use KE = (1/2)I_Dω² when setting up mgh = KE. You can then solve for ω. Then use ω to find V_B.

It is late here (about 1 am) and I do need my beauty rest. So I will quit for now and check back tomorrow. :zzz:

 

[postfan]

Alright, thanks for all your help! You're awesome!

 

[postfan]

TSny,How did you draw the diagrams?

 

[TSny]

I used Microsoft Paint.

 

[postfan]

How did you get the square to rotate?

 

[TSny]

In some cases I just redrew it in a rotated position. Sometimes I copy a drawing that I drew in Paint into another program that will allow me to easily rotate the figure and then copy the rotated figure back into Paint.

For example, Powerpoint allows you to easily rotate a figure any amount you want.