Oscillating Rectangular Plate

In summary, the conversation discusses the suspension of a uniform rectangular plate and the determination of a point along the line between the center of mass and the top center of the rectangle where the plate can be suspended to have the same period of oscillation as it does at the top center. The equation for period is given, as well as the moment of inertia and mass of the plate. The conversation then delves into the algebraic approach to solving for the distance of this point, with some discrepancy between the textbook's answer and the discoverer02's answer. Ultimately, it is determined that the correct answer is x = (a^2+b^2)/6b.
  • #1
discoverer02
138
1
I think I'm lost in the ugly algebra, but I want to make sure.

A uniform rectangular plate is suspended at point P (top center of the rectangle), and swings in the plane of the paper about an axis through P. At what other point between P and O (center of the rectangle), along PO, could the plate be suspended to have the same period of oscillation as it has around P. O is the cm of the plate.
The rectangle has a width and b length.

The answer is (a^2b)/(3(a^2 + b^2))

T = period.
I = moment of inertia
M = mass

1) T = 2pi(I/mgd)^(1/2) I = (1/12)M(a^2 + b^2) + M(b/2)^2

d = b/2

2) T = 2pi[(8b^2 + 2a^2)/(12mgb)]^(1/2)

I then substitute (b/2 - x) for d in equation 1 and set it equal to equation 2 and solve for x, but I'm not getting the correct answer. Is this the correct approach?

Thanks.

The perpetually confused discoverer02.
 
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  • #2
I think you're on the right track, but it's not quite that "easy".

The point you're looking for (let's call it Q) has a different moment of inertia, so the expression for the period when oscillating about Q will be different.

Then, if x is the distance from cm to Q, I think the equation to solve is:
IP/(b/2) = IQ/x

but I haven't been able to get that answer. You're right, the algebra is uuuuugly.
 
  • #3
Where'd you get that answer?

I get
x = (a2+b2)/6b, where x is the distance from the center of mass to the new point Q.

(Assuming my equation IP/(b/2) = IQ/x is correct.)
 
  • #4
The answer came from my trusty textbook.

I did remember to substitute the new distance from cm to pivot point into the moment of inertia, but try as I may I don't come up with the same terms the book does.
 
  • #5
I was basically trying to solve the same equation except I was using b/2-x instead of x which makes the problem even uglier. You're solving for the distance from the pivot and I'm solving for the distance from the center of mass, but all in all both equations seem OK to me. I've posted my question on our class's bulletin board also. I'll let you know if there's a problem with the answer in the book.

Thanks for your help.
 
  • #6
You're welcome.

My answer seems to check out. If x = (a2+b2)/6b, then

IQ = Icm + m((a2+b2)/6b)2
IQ = (m/12)(a2+b2) + (m/36b2)(a2+b2)2

and if you divide that mess by (a2+b2)/6b you get a worse mess that eventually simplifies to
(ma2 + 4mb2)/6b

And similarly
IP = Icm + (mb2)/4
IP = (m/12)(a2+b2) + (mb2)/4
and if you divide that by 2b, it also becomes
(ma2 + 4mb2)/6b

So I think that proves that x = (a2+b2)/6b
 

1. What is an oscillating rectangular plate?

An oscillating rectangular plate is a flat, thin sheet of material that is fixed at its edges and can vibrate or oscillate in a vertical or horizontal direction. It is commonly used in experiments and research to study the effects of vibration on different materials.

2. How does an oscillating rectangular plate work?

An oscillating rectangular plate works by applying a force, such as a mechanical or electrical force, to the center of the plate. This causes the plate to vibrate in a specific mode or pattern, depending on the frequency and amplitude of the force.

3. What are the applications of an oscillating rectangular plate?

Oscillating rectangular plates have various applications in different fields of science and engineering. They are used to study the behavior of materials under vibration, to test the strength and durability of structures, and to develop new technologies such as microscale devices and sensors.

4. How is an oscillating rectangular plate different from other types of oscillators?

An oscillating rectangular plate differs from other types of oscillators, such as pendulums or tuning forks, in its shape and mode of vibration. While other oscillators have a fixed point of rotation or a specific frequency of vibration, an oscillating rectangular plate can have multiple modes of vibration and can be controlled by changing the frequency and amplitude of the applied force.

5. How is an oscillating rectangular plate used in research and experimentation?

In research and experimentation, an oscillating rectangular plate is used to study the effects of vibration on different materials and structures. It can also be used to develop and test new technologies, such as microscale devices and sensors. Researchers can control the frequency and amplitude of the applied force to observe how the material or structure responds, and use this information to make improvements or advancements in their field.

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