ssProblem $11\mathrm{.}24$

In elementary beam theory, for a uniform beam supported as shown, the relation between the force $P$ applied at the end of a beam and

the corresponding end deflection $$ is $P=(3E\frac{\text{I}}{L^3}),$ where $E$ is a constant called the modulus of elasticity, I is a constant called the

centroidal area moment of inertia, and $L$ is the length of the beam. If the dimensions of I are length to the power $4,$ determine the SI

units used to measure the constant $E.$

Problem $11\mathrm{.}26$ i

Docking maneuvers in orbital mechanics can be described using linear orbit theory. This theory provides a simplified description of the

required changes in position and velocity of a vehicle as it approaches and seeks to dock with another vehicle in a reference orbit. When

the reference orbit is elliptical, a quantity $q$ appears in the expression for position change with $q$ given by:

$q=\frac{(\frac{p}{h})(p+r)rsin-3\text{e}\text{p}\text{t}}{1-e^2}$

In this equation, $e$ is a dimensionless measure of the elliptical nature of the orbit and is called the eccentricity The radial position

of the reference orbit, $r,$ is given in units of $km.$ Time $t$ is given in seconds, and $$ is an angular displacement measured from the perigee of

the reference orbit. Find the SI units of $h$ in terms of $km$ and $s$ as well as the corresponding units of $q.$