Friday, January 06, 2017

Friday Gen Eds MS12: The Forces of Motion

This is the twelfth post in the math/science part of my "Gen Eds in a Nutshell" series. The Gen Ed series consists of ten subjects you might study in a general education or "liberal arts" core at a university or college. I've already done the subject of philosophy, and I'm over half way through the world history subject on Wednesdays. I'm combining the last two on math and science into one series on Fridays.

Thus far in the math/science subjects:
Four Fundamental Forces
1. At present, physics talks about four basic forces in nature. The one most familiar to us is gravity, although we will ask in a later post if gravity is really a force or rather the curvature of space around massive objects. Some suggest it is based on an as yet undiscovered particle, the "graviton."

The second force we recognize the most is the electromagnetic force, which is the basis for electricity and magnetism. These are the forces that hold atoms together. These are the forces that cause friction. Ultimately, this force is behind how we move things around or how things react chemically with each other.

The other two forces have to do with the nucleus of atoms and so, while they are absolutely essential to existence, they are completely foreign to our experience. The strong nuclear force is what holds the protons of the nucleus together, even though they have the same charge. The weak nuclear force is what causes a nuclear process by which a neutron decays into a proton. [1]

Newton's Three Laws
2. In the 1600s, Sir Isaac Newton (1643-1727) set forth three "laws" relating to force and motion. We mentioned the first law in the previous entry. A body at rest wants to stay at rest, and a body in motion wants to stay in motion. The reason this does not happen is because of gravity (the things we throw forward fall to the ground and so cannot continue forward) and friction. We mentioned in the previous entry that the reason we fly off a merry-go-round or slide to the side in a car is the fact that our bodies want to stay in motion in the direction they were headed.

3. Newton's second law is often summarized in the formula F = ma. Force equals mass times acceleration. Another way to express the second law is that force is the instantaneous rate of change in the momentum of an object. We know momentum as the fact that our bodies want to stay in motion. If we are running down a hill, we may find it hard to stop because our body has momentum.

Momentum can be expressed by the formula p = mv, momentum equals mass times velocity. Another quantity is known as impulse, which is force multiplied by the amount of time the force is applied.

Another law is the conservation of momentum. The amount of combined momentum before a collision, for example, equals the combined momentum after a collision.

4. Newton's third law is often captured in the statement--"For every action there is an equal and opposite reaction." So when a bug hits the windshield of your car, the force it acts on the windshield equals the force the windshield pushes back on it. But because the car is so much more massive than the bug (F = ma), it experiences a negligible deceleration (negative acceleration). But the bug experiences a massive deceleration with its tiny mass, causing it to go splat.

Although it may be counterintuitive, the floor is pushing back up on us (the "normal" force) to the same extent as our weight is pushing down on the floor. That is why we do not fall through. This might be a good point to mention that weight is a force and is technically different from mass. Mass has to do with how much "stuff" something consists of. Weight, on the other hand, is the force of gravity on that mass.

So our weight will be different on different planets because the force of gravity will be different. This is why we can jump farther distances on the moon. If we substitute g (the acceleration due to gravity) for a in the formula F = ma, we have the formula that our weight = mg.

Newton's Law of Universal Gravitation
5. Newton discovered that every object in the universe exerts a gravitational force on every other object in the universe. In particular, he discovered that this force decreased in relation to the square of the distance between the two objects. And it increased in relation to the two masses times each other. The formula looks like this:

F = Gm1m2/r2

The force of gravity between two bodies equals a constant (the number 6.67 x 10-11) times the product of the two masses, divided by the distance between them squared.

So the reason the earth holds us to the ground is because it is so massive. The moon is less massive, so its gravitational pull on us would be less.

Work, Power, and Kinetic Energy
6. In terms of motion, physics defines work as the distance for which a force is applied, so W = Fd. The standard unit of force is the newton, named for the scientist, so a unit of work might be the "newton-meter," which is also called a "joule" of work.

7. Another distinction often mentioned at this point in your physics journey is the distinction between potential energy and kinetic energy. Potential energy has to do with a situation where work is "waiting" to be done, where energy is on the verge of being set in motion. Let's say I am holding a rock in the air. If I let loose, it will fall. We might say it has a certain potential energy that can easily be put into motion as kinetic energy, the energy of motion. Similarly, if I have a wound up rubber band, it is ready to exchange its potential energy for kinetic energy and work done.

The formula for kinetic energy is 1/2mv2.

The relationship between kinetic energy and work can be expressed in what is called the work-energy theorem: the total work done equals the change in kinetic energy at the beginning and end or Wtot = K2 - K1 = ∆K. This also embodies another law, the law of conservation of energy. The total energy before and after any event or process will always be the same, even though some of the energy is effectively "lost" as heat (see entry covering the second law of thermodynamics).

8. Power is another category in this discussion, often expressed in watts. One watt is one joule of work being done per second. So power is the change in work done per time or P = W/t . Another way to express this is force times velocity.

Rotation and Force

9. The same basic rules apply to rotational motion as to straight line motion. For example, if we substitute the "angular" distance covered (θ) for the straight line distance and we substitute the "angular" velocity (ω) for the straight line velocity, we find the same distance and velocity equations apply to circular motion as to straight line motion. E.g., θ2 = θ1 + ω1t + 1/2at2.

By angular distance, I mean the angle in radians that a point has moved along the circle (in effect, the number of "radiuses" traveled along the circumference of the circle). Angular velocity is usually just the number of radians per second. The formula for the kinetic energy of a rotating body is directly analogous to the formula for straight line motion K = 1/2Iω2, where I is called the "moment of inertia." It is basically a way of expressing how the mass of a body functions in relation to it spinning on some axis. There are formulas for calculating what it would be depending on the shape of the body and where the axis is that you are rotating it around

10. Torque refers to a force exerted on something you are causing to rotate, such as when you exert force on a wrench to try to turn a bolt. The torque is the force you exert times the "lever arm," where the lever arm is the distance between the point of rotation and the "line of action" where you are exerting the force.

Summary of Formulas
  • momentum: p = mv
  • impulse: J = Ft
  • kinetic energy: KE = 1/2mv2
  • work: W = Fd
  • power: P = W/t
Next Week: Math/Science 13: Electromagnetism

[1] In the early history of the universe (about 10-12 seconds after creation), the electromagnetic and weak forces were probably merged in an "electroweak" force. Even earlier than then (10-34 seconds after creation), the electroweak force was probably combined with the strong force in a grand unified force of some kind. The search for this grand unified force continues.

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