Conservation of energy and angular momentum

[pixel01]

Hi all,

There are 2 identical gears which are in the same axis. At first gear #1 rotates at angular velocity w, while gear #2 stays still. Now the gears are attached and rotate at the same angular velocity w'.

Because the angular momentum is conserved so w' = 0.5 w
But then the kinetic energy is not conserved?

 

[Bob_for_short]

Hi all,

There are 2 identical gears which are in the same axis. At first gear #1 rotates at angular velocity w, while gear #2 stays still. Now the gears are attached and rotate at the same angular velocity w'.

Because the angular momentum is conserved so w' = 0.5 w
But then the kinetic energy is not conserved?

No, kinetic energy is not conserved here. It is kind of inelastic collision.

In linear collisions of two equal masses, when they stick together, it means inelastic collision.

A part of the initial energy is spent on heat to stick the bodies together.

 

[BobG]

Hi all,

There are 2 identical gears which are in the same axis. At first gear #1 rotates at angular velocity w, while gear #2 stays still. Now the gears are attached and rotate at the same angular velocity w'.

Because the angular momentum is conserved so w' = 0.5 w
But then the kinetic energy is not conserved?

This is true. Total energy stays constant. It's free to change form from potential to kinetic, or even different types of kinetic energy - the rotational kinetic energy in this example, linear kinetic energy, or internal kinetic energy (i.e. heat). In fact, some of the energy even takes the form of sound.

 

[pixel01]

This is true. Total energy stays constant. It's free to change form from potential to kinetic, or even different types of kinetic energy - the rotational kinetic energy in this example, linear kinetic energy, or internal kinetic energy (i.e. heat). In fact, some of the energy even takes the form of sound.

The thing is the 'lost' energy is exactly 1/2 the total energy !

 

[Bob_for_short]

The thing is the 'lost' energy is exactly 1/2 the total energy !

It's correct for equal gears (or masses).

 

[BobG]

The thing is the 'lost' energy is exactly 1/2 the total energy !

The same thing is true for linear momentum and linear kinetic energy.

If two objects had the same mass, and the first collided with a stationary second object, you'd expect the first object to be stationary while the second moved at the same speed that the first originally had. The fact that they both move at half the speed is a drastically different scenario. Something had to happen for them to stick together.

If you only had one cog on each gear, you'd expect the first gear to transfer momentum to the second and come to a stop; then the second gear to rotate around and transfer momentum back to the first, etc. That would be an interaction that conserved both momentum and kinetic energy.

The reason both are conserved in the one cog example is that you have an opposite and equal reaction every time the gears interact.

 

[I_am_learning]

Since this thread is talking about collisons I would also like to put a question.
In inelastic collisions energy are lost so kinetic Energy isn't conserved. But it is said that momentum is conserved.
Why must it be always the case that objects move after collisions in a way that conserve s the momentum even when the energy needn't be conserved?What I mean, why can't just they come to rest, all of the lost energy coming off as heat!

 

[BobG]

Since this thread is talking about collisons I would also like to put a question.
In inelastic collisions energy are lost so kinetic Energy isn't conserved. But it is said that momentum is conserved.
Why must it be always the case that objects move after collisions in a way that conserve s the momentum even when the energy needn't be conserved?What I mean, why can't just they come to rest, all of the lost energy coming off as heat!

They can come to rest, but something has to absorb the momentum from the object. For example, a 1000kg car is traveling due East 25 meters per second and eventually comes to rest due to friction and air drag - the Earth absorbs that momentum by spinning faster. Just divide 25000 kg-m by the Earth's moment of inertia ([6 x10^24 kg * 6.4 x 10^6 m]/2 ) and you'll know how much faster the Earth has to spin (in radians per second).

The total energy of the object has to be conserved as well. The only difference is that energy can be converted into different forms (kinetic, potential, etc) and momentum can't.