Rectangular Plate SHM Time Period

[Tanya Sharma]

The blue dot was the origin of the coordinate system. with respect to it, the coordinates of the CM were [Lsinθ , -(Lcosθ+a/2)]. That is, the X coordinate is X=Lsinθ, the y coordinate is Y=-Lcosθ-a/2: Lsinθ=X, -Lcosθ=Y+a/2. Square both equations and add them:

X²+(Y+a/2)²=L².

You got the equation of the circle the CM moves along. The centre is (0;-a/2). The radius is L. What was the guesswork?

ehild

Nice and simple. Thanks!

I didn't find the center ,the way you did . When I looked at the coordinates ,it struck to me that if I could shift the origin to (0,-a/2) ,that will do the trick .

That is the guesswork I was talking about .

 

[Tanya Sharma]

The radius of the arc of the CM must equal the radius of the arc of the point of attachment of one of the strings to the plate. Thus, the radius of the arc of the CM must be the same as the length of the string.

Nice!

Thank you very very much TSny and ehild .You both have been simply wonderful

 

[ehild]

Intuition is very important, and you have that ability. But it should be checked and proved by systematic work, which might not be that nice.

I followed the method applied in high-school problems. 'Prove that something is a circle and find the centre and radius'.
TSny-s method is more intuitive and physical : The plate only translates - all its points move along the same curve , the point of any attachment with a string has to move along a circle of radius L, so does the CM.

ehild