A cafeteria tray dispenser supports a stack of trays on a shelf that hangs from

Question

A cafeteria tray dispenser supports a stack of trays on a shelf that hangs from four identical spiral springs under tension, one near each corner of the shelf. Each tray is rectangular, 45.3 cm by 35.6 cm, 0.450 cm thick, and with mass 580 g. (a) Demonstrate that the top tray in the stack can always be at the same height above the floor, however many trays are in the dispenser. (b) Find the spring constant each spring should have for the dispenser to function in this convenient way. (c) Is any piece of data unnecessary for this determination?

Explanation / Answer

Let the equivalent spring constant of the 4 springs together be K Force on the springs (F) = K * compression in spring (x) F=N * 0.580 * 9.8 (where N is no. of trays and 580 is each tray mass in kg) For each tray added the extra compression of spring is : x=0.58* 9.8/K For stack to be at the same height above the floor the compression should be equal to the thicknes of the added tray. 0.58* 9.8/K = 0.45*10-2 =>K= 1263.024 N/m As the springs are in parallel the spring constant of individual springs(k) = K/4 = 315.756 N/m. The length and breadth of the trays is unnecessary. Note:the answers will vary slightly if g=10m/s2 is taken. Hope this helps. Cheers! :)

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