The uniform thin bar pictured below of length L and mass M is in static equilibr

Question

The uniform thin bar pictured below of length L and mass M is in static equilibrium. The upper spring, which is attached to the center of mass of teh bar, has spring constant K and an equilibrium length of 1.5 L. The lower two springs are attached to the ends of the bar, which is level when in equilibrium. Each has an equilibrium length of 2L and spring constant k. The lower points of these two springs are attached 2L apart and symmetrically with respect to the bars center of mass. Find the potential energy stored in the three springs. Note that the figure is very much not drawn to scale and you may not neglect gravity.

Explanation / Answer

as observed frm the data and the diagram, the strings would have been in their equilibrium length if the bar was massless. since the bar has mass , mg has to be balanced, let x1 be the length by which the vertical spring is stretched. let x2 be the length by which each of the lower spring is stretched. then x2 cos(A) = x1 ; where A is the angle which the lower spring makes with vertical axis. also Kx1 + 2kx2 cos(A) = Mg (K+2k) x1 = Mg x1 = Mg/(K+2k); and tan(A) = 1.5L / (2.5L - x1) substitute value of x1 x2= x1 sec(A) thus we found x1 and x2 the total potential energy in springs is ; 1/2[ K(x1^2) + 2k (x2^2) ]

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