Could you do problem $3$a cutting the beam and getting equation from both LHS and RHS

Problem $1$

For the two sets of conventions given below, derive the differential relationships between $M,S,q$

and write the adjust the sign in the relation between $M$ and $$ appropriately.

Convention $1$

Convetnion $2$

Problem $2$

Show that the bending moment diagram along a stretch of beam that is completely unloaded $($no

external load, no reaction$)$ must be a straight line. Show that along a stretch of beam where the

bending moment is zero throughout, then the beam remains straight.

B $-$ Bending moment diagram and deflection by differential equation

Problem $3$

The following beams are all of uniform bending stiffness $EI$ and of length $L.$ For each case:

$($i$)$ Draw the free body diagram.

$($ii$)$ Derive an expression for the bending moment using the convention prescribed

$($Conventions $1$ and $2$ refer to Problem $1$ above$)$ and draw the BM diagram.

$($iii$)$ Derive an expression for the shear force using the convention prescribed in each case

and draw the SF diagram.

$($iv$)$ Identify the maximum bending moment and shear.

$($v$)$ Find the equation of the deflected shape by solving the curvature$-$deflection relationship

and identify the maximum deflection.

$($a$)$ use convention $\frac{?}{{?}_{?}}$

$($c$)$ Use convention $2$

$($b$)$ Use convention $1$

$($d$)$ Use convention $2$

$($e$)$ Use convention $1$