The tray dispenser in your cafeteria has broken and is not repairable. The custodian knows that you are good at design-ing things and asks you to help him build a new dispenser out of spare parts he has on his workbench. The tray dispenser supports a stack of trays on a shelf that is supported by four springs, one at each corner of the shelf. Each tray is rectangu-lar, with dimensions 45.3 cm by 35.6 cm. Each tray is 0.450 cm thick and has a mass of 580 g. The custodian asks you to design a new four-spring dispenser such that when a tray is removed, the dispenser pushes up the remaining stack so that the top tray is at the same position as the just-removed tray was. He has a wide variety of springs that he can use to build the dispenser. Which springs should he use

In this case for the design of the dispenser the four springs are placed in the four corner at the bottom, therefore we can use the translational equilibrium relationship

4 F_e -W = 0

where the elastic force is

F_e = k x

we substitute

4 kx = mg

k = [tex]\frac{mg}{4x}[/tex]

Each tray has a thickness of x = 0.450 cm = 0.450 10⁻² m, this should be the elongation of the spring so that when the tray is in position it will remain fixed.

let's calculate

k = [tex]\frac{0.580 \ 9.8}{4 \ 0.450 \ 10^{-2} }[/tex]

k = 3.1578 10² N / m

k = 316 N / m

therefore you have to find 4 spring with this elastic constant