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arpon
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Homework Statement
Four weightless rods of length ##l## each are connected by hinged joints and form a rhomb (Fig. 48). A hinge A is fixed, and a load is suspended to a hinge C. Hinges D and B are connected by a weightless spring of length ##1.5l## in the undeformed state. In equilibrium, the rods form angles ##\alpha _0 = 30° ## with the vertical. Determine the period T of small oscillations of the load.
Homework Equations
##U = mgh ##
##U = \frac{1}{2} kx^2 ##
The Attempt at a Solution
I used energy conservation law in this case. But this gave me crazy results. Surely, I have made a mistake to apply this law. Would you please help me to find out the mistake ?
So, ## h = 2l cos \alpha## ... (i) ;
##y = 2l sin \alpha## ;
Expansion (or compression) of the spring, ## x = y - 1.5 l = l ( 2sin \alpha - 1.5)## ...(ii)
Let the spring constant be ##k## and the mass of the load be ##m##;
Applying energy conservation law :
## \frac{1}{2} kx^2 - mgh = constant ## [when h increase, gravitational potential decreases]
## kx \frac {dx}{dt} - mg \frac {dh}{dt} = 0##
##kx - mg \frac {dh}{dx} = 0## ... (iii)
But, from eq. (i) & (ii), ## \frac{dh}{dx} = - tan \alpha ##
So, (iii) >>
## kx = -mg tan \alpha ## ;
## k = - \frac{mg tan \alpha}{x} = - \frac{mg tan \alpha}{ l ( 2sin \alpha - 1.5)} ##
So, ##k## becomes variable.