Free help with homework

Why join Brainly?

  • ask questions about your assignment
  • get answers with explanations
  • find similar questions

A square plate of side 'a' oscillates about an axis perpendicular to its plane through a point along 1 of the diagonals distant halfway between 1 corner

& the centre of the square. Calculate its radius of gyration



This Is a Certified Answer

Certified answers contain reliable, trustworthy information vouched for by a hand-picked team of experts. Brainly has millions of high quality answers, all of them carefully moderated by our most trusted community members, but certified answers are the finest of the finest.
See Diagram.

   First we will find the Moment of Inertia of a rectangular sheet/plate of surface density  m units/square area.  Let the total mass be = M = m a b where a and b are the width and height of the plate.    M = 4 m s t.

Given,  P is midpoint of half of diagonal =>  OP = √(a²+b²)  / 4

Moment of Inertia of plate about X axis is :
I_X= \int\limits^{+\frac{b}{2}}_{-\frac{b}{2}} {r_y^2} \, dm \\\\= \int\limits^{\frac{b}{2}}_{-\frac{b}{2}} {y^2} \, ( m\ a\ dy)\\\\=\frac{1}{6}m\ a\ b^3=\frac{1}{6}M\ b^2\\\\Similarly,\ I_Y=\int\limits^{+\frac{a}{2}}_{-\frac{a}{2}} {r_x^2} \, dm \\\\= \int\limits^{\frac{a}{2}}_{-\frac{a}{2}} {x^2} \, (m\ b\ dx)\\\\=\frac{1}{6}m\ b\ a^3=\frac{1}{6}M\ a^2\\\\I_Z=I_X+I_Y=\frac{1}{6}M\ (a^2+b^2)\\\\I_{Z_P}=I_{Z_O}+M*OP^2\\\\=\frac{5}{12}M(a^2+b^2)=\frac{5}{6}M\ a^2,\ for\ a\ Square\\

1 5 1
click on thanks button above pls;;;select best answer
The Brain
  • The Brain
  • Helper
Not sure about the answer?
Learn more with Brainly!
Having trouble with your homework?
Get free help!
  • 80% of questions are answered in under 10 minutes
  • Answers come with explanations, so that you can learn
  • Answer quality is ensured by our experts