3 A block of mass 200 kg is connected to a horizontal ceiling by four identical light elastic ropes, each having natural length 7 m and stiffness \(180 \mathrm {~N} \mathrm {~m} ^ { - 1 }\). It is also connected to the floor by a single light elastic rope having stiffness \(80 \mathrm { Nm } ^ { - 1 }\). Throughout this question you may assume that all five ropes are stretched and vertical, and you may neglect air resistance.
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Fig. 3 shows the block resting in equilibrium, with each of the top ropes having length 10 m and the bottom rope having length 8 m .
- Find the tension in one of the top ropes.
- Find the natural length of the bottom rope.
The block now moves vertically up and down. At time \(t\) seconds, the block is \(x\) metres below its equilibrium position.
- Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 4 x\).
The motion is started by pulling the block down 2.2 m below its equilibrium position and releasing it from rest. The block then executes simple harmonic motion with amplitude 2.2 m .
- Find the maximum magnitude of the acceleration of the block.
- Find the speed of the block when it has travelled 3.8 m from its starting point.
- Find the distance travelled by the block in the first 5 s .