Conservation Laws: Energy, Momentum, and Angular Momentum.

 

 

Force Things First....

To begin our study of conservation laws, lets review Newton's three laws, and some QuickTime movies that illustrate them

 

  1. Acceleration = Net Force/ Total Mass (Newton's second law)

     

    Movie 1 shows a mass moving on a frictionless surface. It is acted upon by constant 10 N force. Note that the acceleration is constant, and the velocity increases as time goes on.

    Movie 2 shows a mass thrown into the air. Note that gravity is always acting on the mass, the acceleration is always constant and downward. The velocity decreases, goes to zero, and then increases in the downwards direction…

  1. Objects keep moving with constant velocity unless a net force acts on them
  2. Movie 2 shows what happens when the net force gets smaller and finally reaches zero. This time a mass moving on a frictionless surface with the same applied force, but there is air resistance. Since the force of air resistance increases as the speed gets greater, the net force (10 N minus the applied force) gets less and less until the acceleration equals zero. At that point, the speed is constant. How do you think this relates to the concept of terminal velocity for skydivers?

     

  3. If object 1 applies a force to object 2, then object 2 applies an equal and opposite force to object 1.

Check out this (not available yet...check back soon)movie by Doug Brown at Cabrillo College, showing two ice skaters pushing on one another. What do you think would happen if one ice skater was 10 times as massive as the other?

 

 

Answer: the lighter skater would accelerate 10 times as much as the heavier skater, but they would both move away from their center of mass...

Back to Work:

You will recall that Work is Force x distance. Imagine that the force is you lifting your physics book. As before, the force you need to apply equals the weight of your book. Also imagine that instead of gravity, there is a spring attaching your book to the ground. As you lift the book, you stretch the spring. Now, the work you do ends up as Potential energy stored in the spring. When you let go of the book, the spring stretches back and the potential energy gradually turns into kinetic energy until the book is nearly back on the floor, at which point all the potential energy has been converted to kinetic energy.

How does this relate to gravity? Well, when I lift something, I think about stretching a "gravitational spring". The work that I do ends up as potential energy stored in the Earth's gravitational field, as if the field is a bunch of springs attaching the Earth to objects above its surface.

image of gravitational spring and upward movement is applied

When the object is lifted to its final height (h), the work you've done (mgh) is now stored as "potential energy" in the Earth's gravitational field.

image of gravitational spring

You can think of the field as being like "gravitational springs." Also like springs, when you let go, the gravitational field pulls back doing work on the object, giving the object "kinetic energy". Just before the object hits the ground, the Kinetic energy equals the potential energy.

As you can read in the text, the amount of kinetic energy is determined by the speed and mass of the object:

 

 

Thus if you want to calculate how fast an object will hit the ground, set the potential energy equal to the kinetic energy:

 

Using numbers: if h = 20 m, then

V = 20 m/s (try it!). (Of course, you could get the same result using free fall formulas! From free fall, h = 1/2 gt2 you get t = 2 seconds, and using v = gt, you get

V = 10m/s2 x 2 seconds = 20 m/s. Done be alarmed if there are more than one way to do something -- physics provides many different tools: the trick is to know which one is best! )

 

Whatever can happen will happen…

Lets change gears for a moment and consider a toy that most of you have seen: the old swinging balls apparatus:

 

If you pull one ball out and let it go, what happens? Try it over and over until your fingers are numb and you will find that one ball always comes out the other side. Never two or three, always one! Is Energy conservation the reason this happens?. Consider for a second what would happen if two balls came out instead of one. Since the kinetic energy would be the same before and after the collisions, you could write:

Kinetic energy in = Kinetic Energy out

If one ball has Mass m and speed v1, then two balls, with mass 2m and speed v2 would come out. By conservation of energy:

image of kinetic energy in and kinetic energy out

So if energy conservation were the only law governing the balls, then two balls could come flying out, instead of one, as long as they were moving a bit more slowly (and would rise up as high). But since this never happens, perhaps another law is governing this game! (also note that not all the kinetic energy of one ball is transferred to the other…you do hear a "clack" sound when the balls hit. That sound is energy passing through the air and reaching your ear, and it came from the energy of the first ball!)

What about Momentum

If you are keeping up in your reading, then you know that Momentum is also a conserved quantity. When two objects collide they exert equal and opposite forces on one another, and so their impulses are equal and opposite, and the change and momentum are equal and opposite:

F1 = - F2

Dp1= -Dp2

meaning that whatever momentum one looses, the other one gains. Check out this movie to see momentum conservation in action.

Getting back to the balls: Since momentum is mass x velocity, or mv, we can look at the collisions in terms of momentum conservation.

Momentum in = Momentum out

Lets consider the case where one ball goes in, and two balls come out. If the masses and velocities use the same symbols as before:

mv1 = 2mv2

This means that that v2 = v1/2

Thus, according to momentum conservation, one ball could go in, and two balls could come out, so long as the two balls had half the velocity of the one ball. However, energy conservation said that two balls should have a velocity that is equal to the velocity of one ball divided by the square root of two, meaning that the two laws predict two different results…how is this possible?

The answer is that it isn't possible because it won't happen. In order for something to happen, it must satisfy all the conservation laws, not just one or two. The only way to satisfy both momentum and energy conservation in this case is for one ball to go in and one ball to go out, or two balls in, two balls out!

image of two balls on the swinging balls device swinging from theleft  before they strike the three stationary balls

image of balls on the swinging balls device swinging right to the right

 

Note that Conservation of Momentum doesn't apply to only straight line motion. Below is a strobe photo of two pucks colliding. The red put is initially is at rest, and the green puck is fired at it. After the collision the red and green pucks go off at separate angles.

image of  two pucks colliding

Imagine for a moment that the red puck was invisible on the photo. You could infer its existence by the path of the visible green puck. In fact, from the laws of momentum and energy conservation, you could determine the mass and speed of the red puck, even though you couldn't see it. This is exactly what physicists do when analyzing photos if collisions among elementary particles. Below is a "bubble chamber" photo from Lawrence Berkley lab. By applying the laws of energy and momentum conservation (along with a few other laws we'll talk about later) physicists have been able to identify a large number of particles…most of which leave no track! The "interaction point" is at middle right in the photo. The circles are due to particles spiraling in the magnetic field of the chamber.

image of particle spiraling

 

 

Other conserved quantities:

Besides momentum and energy, Angular momentum, and Charge are also conserved in "interactions" Angular momentum is the product of an objects moment of inertia and its rate of rotation. Just as the momentum of interacting objects is constant as long an no external force acts on them, the angular momentum of one or more objects is constant as long as no external torque acts on them (see the text!)…Think of an ice skater who pulls in her arms while spinning. Her rate of rotation increases as her moment of inertia decreases. You can experiment with this by visiting a playground. Find a carousel or a hanging tire, and get yourself moving in a circle. Now move your arms in or out, or move closer to the center and watch the spin rate change. You can also do this on a rotating chair (just be sure the boss isn't watching).

One of the best examples of angular momentum conservation is the Earth!:

Without any significant "torques" to slow it down, the Earth keeps spinning at the same rate, year after year, month after month. Also, since angular momentum is a vector, the polar axis points in the same direction as the Earth orbits the sun. However, the moon exerts a tiny torques on the Earth. The Earth's spin slows slightly as a result (about 1ms/century). However, since the angular momentum of the Earth and moon system is constant, the moon moves farther away from the Earth (by 1cm/year)--increasing its moment of inertia to compensate for the loss of angular momentum of the Earth. Did you know that the day was slowing down? It sure seems just the opposite to me!