Question
Asked Dec 16, 2019
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Two springs of force constants k1 and k2 are attached to a block of mass m and to fixed supports as shown in Fig. 2. The table surface is frictionless.

(a)  If the block is displaced from its equilibrium position and is then released, show that its motion will be simple harmonic with angular frequency w = √(k1 + k2)/m.

(b)  Suppose the system is now submersed in a liquid with damping coefficient b , what is the condition that the block will return to its
equilibrium position without oscillation?
Sketch a graph to show the behavior of the system in this case.

k1
k2
Fig. 2
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k1 k2 Fig. 2

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Expert Answer

Step 1
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The mass of the block is m and the Spoing constants of both spaing i Ki and kz. Part (a) when the block i moved n distante toward rignt fom the eguillibafum þesition, the pigrt spring Lompressed by distance x wih forre Fi=akn , whereas the left spoing expand by distance a and it pull the block with fosce, Fz and push the block

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So net forre Fit Fz - -k,x-kn on the bloch 'net = -(k+k2)a Ta kfng Fret = - knep r where Knet is the effective Apafng Lonstant --(ktkz)n - kner a Isnet KitKz

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Sping is attached to the block of mass In pasallel, So the net sping Constant is knet k, tkz. The differential equation of clamped free vibratim is knet' (Kae)a x=D0 m. knet Angulor frequency, Wo= So, Kit k2

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Vibrations

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