Angular Acceleration for Angular Pressure of 1.5atm

[fatcat39]

Someone actually posted this exact problem 4 months ago, but here it is. There wasn't an answer.

Homework Statement

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.

Homework Equations

The Attempt at a Solution

Okay, so the moment of inertia is just (2/3)MR^2, I think, because the two rods are essentially massless and have no bearing on I, right?

B) Pressure is Force/Area, so therefore, Force = Pressure * Area. The pressure is 1.5 atm - 1 atm, so 0.5 atm. But is the area the radius of the two spouts? or the volume of the entire sphere?

Thanks in advance.

 

[learningphysics]

The force through each spout is (1.5atm - 1atm)*(area of one spout).

So you have two forces (same magnitude)...

You can then find the net torque about the point of contact with the ground (so this way you don't have to worry about friction with the ground)... then torque = I*angular acceleration... (where I is the moment of inertia about the point of contact... not about the center of mass of the sphere).

here you're finding the angular acceleration of the sphere about the point of contact... but this equals the angular acceleration about the center of mass... so this is the value you need.

 

[ogb p]

I would approach this problem by first defining the variables and units involved. The pressure given is in atmospheres (atm), the radius of the spherical container is in meters (m), and the mass is in kilograms (kg). I would also note that the pressure given is the difference between the inside and outside pressure, so we can calculate the force using only 1.5 atm.

Next, I would use the formula for torque, which is torque = force * distance. In this case, the force is the pressure multiplied by the area of the spouts, and the distance is the radius of the spherical container plus the additional 0.1 m from the spouts. This would give us the torque generated by the steam.

To calculate the angular acceleration, we can use the formula for angular acceleration, which is angular acceleration = torque / moment of inertia. The moment of inertia for a hollow sphere is (2/3)MR^2, where M is the mass and R is the radius. We can calculate the moment of inertia for the spherical container using this formula and the given values.

As for the area, it would be the area of the spouts, since that is where the force is being applied. We can calculate the area of each spout using the formula for the area of a circle, A = πr^2, where r is the radius of the spout (0.01 m).

So, in summary, to calculate the angular acceleration, we need to calculate the torque generated by the steam, the moment of inertia of the spherical container, and the area of the spouts. Once we have these values, we can plug them into the formula for angular acceleration and solve for the answer.

I hope this helps and provides a clear and scientific approach to solving this problem.