Question

An instrument package of mass M is attached to the wall of a moving vehicle by two identical thin rigid beams of mass m. Assume small angular motions of the supporting beams.

Answer 1

The natural frequency ω of the instrument package attached to the moving vehicle, considering small angular motions of the supporting beams, is given by the formula ω =
`√((2g/R))`, where g is the acceleration due to gravity and R is the distance between the attachment points on the wall.

Step-by-step explanation:

In this scenario, we consider the dynamics of the instrument package attached to a moving vehicle by two identical thin rigid beams. Assuming small angular motions of the supporting beams, we can analyze the system's behavior in a simplified manner. The key factor influencing the natural frequency ω is the effective length of the supporting beams.

To derive the formula, we start by considering the rotational motion of the beams. The moment of inertia I of each beam can be expressed as I = (1/3)m
`L^2`, where m is the mass of each beam and L is the length. The total moment of inertia
`I_(total)` for the system is then 2I. Applying the parallel axis theorem, the effective moment of inertia
`L_(eff)` becomes (5/3)m
`L^2`.

Using the formula for the natural frequency of a simple pendulum ω =
`√((g/L_eff))`, where
`L_(eff)` is the effective length, we substitute I_eff to get ω =
`√((3g/5L))`.

Now, considering the attachment points on the wall and the geometry of the system, the effective length L_eff is related to the distance R between attachment points as
`L_(eff)` = R/2. Substituting this into the previous expression, we arrive at the final formula for the natural frequency ω =
`\sqrt(2g/R)}`.

This expression captures the relationship between the acceleration due to gravity, the distance between attachment points, and the natural frequency of the instrument package in small angular motions.

The complete question is:

An instrument package of mass
`\( M \)` is attached to the wall of a moving vehicle by two identical thin rigid beams of mass
`\( m \)`. Assume small angular motions of the supporting beams. If the distance between the attachment points on the wall is
`\( R \)`, derive the formula for the natural frequency
`\( \omega \)` of the instrument package in terms of the acceleration due to gravity
`\( g \)` and the given parameters. Consider the moment of inertia of each beam and the effective length of the supporting beams in your analysis.