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# Oscillations of mass suspended by a spring

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Two small blocks, A and B , of masses 0.8 kg and 1.2 kg respectively, are stuck together. A spring has natural length 0.5 meters and stiffness of 98 N/m. One end of the spring is attached to the top of the block A and the other end of the spring is attached to a fixed point O.

(a) The system hangs in equilibrium with the blocks stuck together, as shown in the diagram. Find the extension of the spring.

(b) Show that the potential energy of the spring when the system is in equilibrium is 1.96 J.

(c) The system is hanging in this equilibrium position when block B falls off and block A begins to move vertically upwards. Block A next comes to rest momentarily when the spring is compressed by x metres. (It will then oscillate up and down at the end of the spring stopping momentarily every half cycle but that does not concern us here.)

(i) Show that x satisfies the equation x^2 + 0.16x - 0.008 = 0

Hint: This is an energy question. The energy of block A and the spring at the moment when block B falls off is the same as the energy of block A and the spring when block A next comes to rest.

(ii) Find the value of x.