The discussion of a violation of the conservation of energy or angular momentum perked up my curiosity. I have searched many times for a trick that gets around the rules to no avail. So, I decided to simulate a simple yet elegant model system that I think might be of interest to the group. My model consists of a mass attached to a lossy spring which is attached to a stationary rod. The heating due to the conversion of spring energy is collected within a thermal storage mass. I used OpenModelica as a platform and suggest that everyone obtain a copy of this very powerful modeling program. For initial conditions I gave the 1kg mass a velocity of 1 meter per second in the X direction. The spring, which has a spring constant of 1 Newton per meter is stretched by 2 meters to reach the mass located along the Y axis. The spring loss, or damping constant was chosen to be .2 n*s/m so that the Moment of Inertia changes gradually as the orbit of the mass around the fixed point slowly changes from what appears to be an ellipse to a circle as the system reaches a stable condition. The initial system contains 2.5 joules of energy. Two joules are contained within the stretched spring according to the equation E=1/2*K*Y where K=1 and Y=2 meters. One Half joules are in the form of kinetic energy of the mass which is 1 kg moving at a velocity of 1 meter per second. My thermal energy capacitor begins at 0 Kelvin with no heat storage. When all the transient motion had settled down I then calculated the total energy to be the same 2.5 joules as during the initial phase. At the end of the simulation the spring was only stretched to the square root of two long from a beginning of 2 meters. That meant its energy was only 1 joule after simulation. The mass is now moving at a velocity of the square root of 2 in a circle so it has increased its energy to 1 joule. Finally, the heat storage capacitor reads .5 degrees Kelvin which translates into .5 joules of energy. As you can see, the energy of the closed system remains exactly the same while the Moment of Inertia is allowed to change in a smooth manner. Of course, the angular momentum is shown to be constant as theory would require. The initial angular momentum of the mass is calculated by M=angular velocity*Moment of Inertia. Angular velocity is Tangential velocity / radius or .5 in the initial case. The Moment of Inertia is M*R squared or 1*2 squared which is 4. So total angular momentum is .5 * 4 = 2. When the dust settles and the simulation completes you will see that the orbit is circular and has a reduced radius of the square root of two meters compared to an initial value of 2 meters. The mass has a linear velocity of the square root of two meters per second which is increased from its original value of 1 meter per second. The final Moment of Inertia has been reduced to the square root of two squared or a value of two. The final angular velocity of the mass becomes the square root of two meters per second divided by the radius which is also the square root of two. The angular velocity yields 1 radian per second. Taking the product of the angular velocity and the MoI gives you the angular momentum of 2*1 or 2. I am confident that everyone noticed that my model consisted of 1 small mass attached to a spring which is held fixed at its far end. The actual complete system has an identical mass and spring connected to this point and symmetrically opposite in position and velocity, etc. So, think of two masses connected at opposite ends of a lossy spring. They are rotating about a common point that is fixed in space with no linear motion. I hope that this simple model will inspire others to play around with mechanical system modelling to obtain a better understanding of physics.
Dave Sent from Mail for Windows 10
