A uniform table of mas M stays horizontally and symmetrically on two wheels rotatig in opposite directions figure. The separsation between the wheels is L. The friction coefficeint between each whee, and the plate is mu. Find the time period of oscilationof the pate if it is slightly displaced along its length and released.

A uniform table of mas M stays horizontally and symmetrically on two w

1.	A uniform table of mas M stays horizontally and symmetrically on two wheels rotatig in opposite directions figure. The separsation between the wheels is L. The friction coefficeint between each whee, and the plate is mu. Find the time period of oscilationof the pate if it is slightly displaced along its length and released.
Answer» SOLUTION :Let X be the displacement of the PLANK TOWARDS left. Now the centre of gravity is also displaced through x. `R_1+R_2=mg` Taking moment about g we get `R_1(l/2-x)=R_2(1/2-x)` `=(mg-R_1)(l/2+x)…………1 `ltbr.gt `S, R_1(l/2-x)=(mg-R_1)(l/2+x)` `rarr R_1 1/2-R_1x=` `mgl/2-R_1x+mgx-R_1l/2` L `rarrR-1l/2+R_1l/2=mg(x+1/2)` `rarr R_1(l/2+l/2)=mg((2x+1)/2)` `rarr R_1l=mg(2x+1) R_1=mg((1+2x)/(2l)` .........i Now, `F_1=muR_1=mumg((1+2x))/(2l)` similarly `F_2=muR=mumg((1+2x))/(2l)` similarlty `F_2=muR_2=muMg((1+2x))/(2l)` SIN `F_1gtF_2` `rarrF_1-F_2=ma-(2mu(mg)/l)x` `a/x=2mu=(mg)/l=omega^2` `rarr =omegamusqrt(g/l)` `: Time period `=2pisqrt(l/(2mug))`