How Does Heron's Ancient Steam Engine Calculate Angular Acceleration?

[Phoenix314]

Hi,

I came across this problem when browsing through a AP Physics B book (the answer was not in there). Can anyone help me?

Heron of Alexandria invented the steam jet engine in the first century A.D. One of his many inventions, the one shown below was invented for amusement but employs many concepts not again used until the 18th century.
A caldron with water in it was heated by fire and the steam generated was fed up and into a hollow spherical container with two spouts on each side. The exiting steam would spin the container at high speeds. We want to estimate the highest rotational speed using the little facts we have about this ingenious device developed almost two thousand years before it was rediscovered as the steam engine.
The spherical container has a radius of 0.2 m and mass of 10 kg. The two spouts can be considered massless but extend an additional 0.1 m above the surface of the container. The container is hollow and do not consider the moment of inertia of the steam contained inside.

b) If the pressure inside of the container reaches 1.5 atmospheres what is the angular acceleration of the container? Take the area of the spouts to be circles of radius 0.01 m. (Remember that the outside pressure is 1 atmosphere).

I know Force = Pressure*Area, but since steam is coming out of the container to push the sphere around, should I calculate force as Force = (1.5 atm + 1atm) * area or just Force = 1.5atm * area. Then from there, I calculate torque and moment of inertia to calculate angular acceleration.

The second problem

Gyroscopes are used in spacecraft and aircraft as navigational aides. A gyroscope is usually mounted on gimbal mounts that allows it to rotate in any direction. Explain why you think this would make a good navigational device. What is the best shape for the spinning object at the center of the device, why?

I've heard that the gyroscope just acts like a compass pointing in one direction, except it doesn't need to be magnetized. Is that all?

(The attachment has the original problems with a few pictures)

 

[abercrombiems02]

For your gyroscope question, the answer is because it acts like a 3-D compass. Standard compasses use magnetic fields to align themselves. Well gyroscopes work the same way but without magnets. Gyroscopes use the concept of angular momeum. As long as no external torque is applied to a rotating body, the angular momentum (which is a vector) is constant. And any change in an objects orientation can be compared to it original orientation in which the gyroscope will be initially spinning with respect to.

 

[trytohelp]

Your force equation should read "Force= .5atm.*area" since the 1 atm.s' cancel each other out. The only active part is the .5 atm.

 

[Phoenix314]

Thanks, I solved the steam engine one and got (pi*10^-5)/0.27 as the angular acceleration