Consider an electrical switch which is modelled as a cantilever beam of length L with a uniformly distributed load q0. The deflection and slope of the beam are zero at point A, while point B can move downwards until it makes contact and completes the circuit.

To keep the switch in equilibrium, the fixed support at A provides the vertical reaction force VA = Lq0, (1)

and the moment

MA = (( L^2)/2)*q0, (2)

Under these external forces, there exist forces inside the switch, namely the shear force, denoted by V , and the bending moment, denoted by M. It is noted that V (x) = dM(x)/dx, where 0 x L. The internal bending moment M can be mathematically modelled as

d^2M(x)/dx^2 = q(x), (3)

which is subject to the initial values

M(0) = MA, (4) V (0) = VA. (5)

The deflection y of the beam can be mathematically modelled as

d^2y(x)/dx^2 = M(x) EI , (6)

which is subject to the initial values y(0) = 0, (7)

dy/dx(0) = 0. (8)

In (6), E (the modulus of elasticity) and I (the second moment of area) are the material and geometric properties of the beam, respectively. In this assignment, the cross section of the beam is of circular shape of radius r, and the following values, which are not to be shared with anyone else, are to be used

L = 10 cm, (9)

E = 60 GPa, (10)

I = 0.30r4 , (11)

r = 5 mm, (12)

q0 = 60N/m. (13)

Your tasks are to numerically solve the initial-value problem (IVP), defined by (3)-(5), for the solutions M(x) and V (x), and then the IVP, defined by (6)-(8), for the solutions y(x) and dy(x)/dx.

Derive the analytical solutions M(x) and V (x) of the IVP (3)-(5), and the analytical solutions y(x) and dy(x)/dx of the IVP (6)-(8).