jrm2002 wrote:According to Newton's second law if the resultant force acting on a body is non zero the body moves with an acceleration proportional to the mass of the body, right?Then,
1. Can anyone explain this in an intuitive sense?
2. In solving problems we say that the resultant force is equivalent to the product "ma"(mass times acceleration).How?
My attempt to explain this in english.
1 and 2, the equation is: F=MA
Force as useful concept is a pretty modern idea. What we're talking about here is almost a dictionary definition of "force" expressed in mathematical terms. To understand it, you have to understand what he means by mass and accelleration.
Weight is something you understand - we all know what we weigh and what heavier and lighter means. The concept of weight as we know it intuitively though requires gravity. We experience weight because the earth's gravity is pulling us downward. On a lighter gravity planet, like the Moon or Mars, you would weigh less. But...your MASS would be the same. Mass is the part of your weight that does not depend on outside forces like gravity. It is the same no matter where you stand.
Accelleration is a concept that is probably easier for us to understand today, with automobiles and such, than it would be for the average person in Newton's time. When you press the accellerator in a car, the car accellerates
When you press the brakes, it "accellerates" in the opposite direction (slows down). Accelleration can be thought of as a measure of whether you are speeding up or slowing down. Zero accelleration means you maintain the same speed (example, in a car, you are at zero accelleration when you are at rest, or when the cruise control is set to 50mph and you stay at that speed)
If you drop a rock, it accellerates toward the earth. It goes from zero speed to some faster speed...that means it is accellerating.
What Newton said with this law is that the rock has a mass, and if we observe how it accellerates we can measure the gravitional "force".
Or if we know the gravitational force, we can take the weight and determine the mass and predict how it will accellerate. Furthermore we can predict how it will behave if some other force (for example, the energy from a bowstring or cannon blast) operates on the object.
You can take any two things in the equation and predict the third. Some are easier to measure than others, depending on your situation. If you have the right kind of math (either really complex geometry or relatively simple Calculus - a math invented by both Newton and a German rival about the same time) you can pretty much predict the way objects behave with all sorts of forces acting on them, if you can figure out how to measure either their mass, their speed, their accelleration or the forces acting on them.
The reference to "Vectors" in the expanded version of his second law is the point that forces have a direction. Going back to the automobile example, when you are moving at 50mph south, pushing on the accellerator will cause a force in the "south" direction, where pushing on the breaks will cause a force in the "north" direction.
Prior to Newton, it was assumed that an object in motion would eventually slow down and stop. This concept of "impetus" matches some observations, such as the fact that if you don't touch accellerator or breaks on your car (or use cruise control), it will slow down from 50mph. The reason though, isn't that Newton is incorrect. The reason is that at all times, an automobile going south has some "north" forces on it if it is moving (air resistance being the big one, but friction also playing a part). What cruise control does is add some "south" forces in the form of engine power to counter the "north" forces and keep you going 50mph south.
Aristotilian mechanics matched some observations but didn't attempt to explain them, and didn't match many other observations. Newton's F=MA approach works on almost everything "human" scale. (very small things, very fast things and very massive things behave differently for reasons way too complex to enter into here)
In what my old instructor called the "never never land of physics" (an environment quite similar to deep space, actually) that car wouldn't have any friction, no air resistance and thus if going 50mph, it would stay at 50mph forever, until some other force made it speed up or slow down.
jrm2002 wrote:
3. Can anyone explain the D'Alembert's principle?
I would call this a "trick" actually. This sort of thing is common in real world math, mechanics and electronics - you can't solve a problem the normal way, so you invent a condition that if it existed, you could solve the problem. Then you kind of back the condition out.
In this case, what D'Alembert is doing is turning a motion problem into a static problem, even when forces are involved, using Newton's equation. This was important to unify thinking, and as a test of the idea. The equation is complicated but all it is really saying is:
"I can prevent accelleration even when forces are operating on an object by applying an opposite force to the other side...an object can hold still because NO forces are applied to it...or because MANY but CANCELLING forces are applied to it"
In fact, Cruise Control and your parking brake on a car can be both seen as examples of the principle in action. The concept is that two opposite forces applied to a given object leave it in equilibrium (not accellerating, changing speeds).
In the 50mph south direction case, we have an existing velocity. The goal of the cruise control is to keep it at 50mph. The road conditions change (wind can help or hinder the speed, hills can help or hinder, plus the car engine/tires will waste some energy in friction) so what the cruise control does is try to exert a force equal to that exerted by the entire environment "against" 50mph. It isn't perfect, because changes in conditions aren't detected until the speedometer shifts higher or lower.
In the parking situation, a car parked on a hill will roll down unless something stops it. If you don't turn your wheels into a curb, or put down some other barrier to it moving, it will accellerate due to gravity. What the parking brake does is increase the force required to cause the tires in the car to rotate (generally, they increase friction).
The difference is that in the first case, going faster or slower is bad, and the environment constantly changes, so you need a way to add "north" and "south" accelleration to the car (brakes and accellearator) and also something smart enough to know when to apply that accelleration.
In the second case, all you want is "don't go faster than zero". So the parking brake only has to be "at least as strong as the force exerted by gravity on any reasonable hill, based on the mass of the car". Unlike with the cruise control, you can make the parking brake way stronger than the expected forces and still get a good result, because gravity isn't going to change. A parking brake though, might not stop the car if an earthquake made the hill steeper. It might also fail if the assumption that the wheels had to turn to move the car failed (if the car can slide on the wheels, like on black ice).