Question

**Angular Impulse and Momentum Principles**

To apply the principle of angular impulse and momentum to describe a particle's motion.

The moment of a force about a point *O*, fixed in an inertial coordinate system, **M***O*, and the angular momentum about the same point, **H***O*, are related as follows:

∑**M***O*=**H**˙*O*

where **H**˙*O* is the time derivative of the angular momentum, **H***O*=**r**×*m***v**. Integrating this equation with respect to time yields the following equation:

∑∫*t*2*t*1**M***O* *dt*=(**H***O*)2−(**H***O*)1

This equation is the principle of angular impulse and momentum, and it is often rearranged to its more familiar form

(**H***O*)1+∑∫*t*2*t*1**M***O* *dt*=(**H***O*)2

A centrifugal governor consists of a central rotating shaft that has two thin, pin-connected rods attached to it; a heavy sphere caps the end of each rod. (Figure 1) A centrifugal governor mechanically limits an engine's speed. A part of the engine turns the centrifugal governor, and if the speed exceeds a set amount, the height of the spheres decreases the driving force of the engine by reducing the fuel flow. The two rods move freely about the pin. If the whole apparatus is rotating about the central shaft and the spheres have a tangential velocity, *v*, the thin rods will create an angle, *θ*, between each rod and the central shaft. Develop an equation for the tangential velocity, *v*, in terms of some or all of the following: *θ*, the angle between the thin rods and the central shaft; *l*, the length from the pin to each sphere's center; *m*, each sphere's mass; and *g*, the acceleration due to gravity. Neglect the mass of the thin rods.Express your answer in terms of some or all of the variables *θ*, *l*, *m*, and *g*.

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Engineering

Mechanical Engineering

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