Class Exercise $23$

Assume you are working as a mechanical engineer for an engineering consulting agency, and in

a design project, you are assigned to design and develop a mechanism for parcel processing for Australian

Post. As shown in the figure below, the objective of the designed mechanism i$.$e$.$ the cradle is to capture

parcels at A and deliver them at $B.$ The cradle is attached to a spring, which allows the cradle to slide on the

hemispherical cylindrical surface. One of the design criteria is that the landing speed of the cradle at $B$ must

not exceed $1\mathrm{.}5\frac{m}{s}.$ Assume the parcels have masses ranging between $5$ and $10$ kg incoming at speeds

between $3$ and $6\frac{m}{s}.$ Compute the acceptable range of

a$)$ the spring constant $k,$

b$)$ the free length of the spring,

c$)$ the kinetic friction coefficient between the cradle and the sliding surface to achieve the desired

results, and

d$)$ discuss and verify the above results whether they are reasonable.

You can use Excel or Matlab tool to numerically solve the value of the parameters. You may use Euler's

method to solve ordinary differential equation. However you must show the logical steps and formulations

to demonstrate the modelling and analysis of the results.