Problem $3$: Compound pendulum attached

to cart

A rigid rod of length $L,$ mass $M$ and centroidal moment of inertia $I_G$ hangs from a pivot as shown. The

angular displacement of the rod is $.$ The base of the rod is attached to a point mass $m$ via a spring of stiffness

$k.$ The point mass is free to roll along a frictionless surface with a displacement $x.$ The spring is unstretched

when $x=0,$ and the entire system is subject to the influence of gravity. Please answer the following questions.

a$)$ Determine the potential and kinetic energies of the system.

b$)$ Determine the equation of motion governing $x.$

c$)$ Determine the equation of motion governing $.$

P$1-$ Singularity Functions

The beam shown in Fig $1$ is has the FBD shown in Fig $2.$

a$)$ Use Singularity function to generate the moment diagram. Show the associated load function $q(x),$ shear

function $V(x),$ and bending moment function $M(x),$ all in terms of singularity functions.

b$)$ Evaluate $V(x)$ and $M(x)$ at $x=0,x=4$ in$,x=8$ in$,8=12$ in and $x=16$ in$.($don$\text{'}$t forget units$)($by hand$)$

c$)$ Create an Excel spreadsheet to create a shear and moment diagram. Use singularity functions. See Lecture $3$

folder on D$2$L for a tutorial on how to create singularity functions using excel. $($paste a picture of our shear and

moment diagrams, follow them by a picture of your spreadsheet$)$

E

Fiaiura $1$

Figure $2$