You will design, build and test a device to measure the mass moment of inertia of a

complex shaped object. The device consists of a $$trifilar pendulum$($a platform suspended from three cables$),$ which is used to hold the test object. The mass moment of inertia of the object can be determined by measuring the vibration frequency of the pendulum. Your design process will involve $($i$)$ Deriving the formula that relates the vibration frequency to the mass moment of inertia; $($ii$)$ Designing the pendulum itself; $($iii$)$ Constructing the pendulum; $($iv$)$ Testing the accuracy of your design; and $($v$)$ using the design to measure mass moments of inertia of several objects with complex system. Here, you will design a device to measure I and use the device to test a range of objects. The device to be designed is the $$Trifilar Pendulum$,$ shown in the figure below. The trifilar pendulum consists of a platform with mass m$0$ and moment of inertia I platform about its center of mass axis $($CM$).$ Three lightweight, uniformly spaced cables suspend this platform a distance L below a rigid support $($not shown$).$ Each of the cables is attached to

the platform at a distance R from the platform CM$.$ The object to be measured is placed on the platform with its center of mass directly above the CM of the platform. The platform is given a small initial rotation and released. The platform oscillates, and has a period $.$ Measurement of the period can be used to deduce the total moment of inertia of the system $($platform plus object$).$

Before starting this problem, notice the following

$($i$)$ The table is rotating about its center, without lateral motion

$($ii$)$ If you look closely at the platform, you will see that it moves up and down by a

very small distance. The platform is at its lowest position when the cables are

vertical.

$($a$)$ The figure above shows the system in its static equilibrium position. The three cables are vertical, and all have length L$.$ The platform has radius R$.$ Take the origin at the center of the disk in the static equilibrium configuration, and let $\{$i$,$j$,$k$\}$ be a Cartesian basis as shown in the picture. Write down the position vectors rD$,$rE $,$rF of the three attachment points in terms of R and L$.$