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Newtons laws

The original translations of newtons laws are

1. Every body continues in its state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces impressed upon it.

2. The change of motion of an object is proportional to the force impressed; and is made in the direction of the straight line in which the force is impressed.

3. To every action there is always opposed an equal reaction; or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.

Modern interpretations phrase them in many different ways, my preferred notation with explanations are

1. If the net force on a body is zero, its velocity continues at a constant (including zero) magnitude and direction.

This law I consider to be redundant to the 2nd law: as for nonzero mass, if force is zero, so is acceleration​

for questions of what laws apply, the 1st is applied for static systems

2. Force is equal to the time rate of change of linear momentum​

in vector form with calculus notation ​

If you're not comfortable with vector notation, in this case, all it means is that the equation is held independently for the x y, and z directions (see the cannonball off a cliff problem). In fact, any cartesian system works, it's different for circular cordinate systems).

For most physics 1 and 2 classes the full equation is overkill, and mass is considered constant with respect to time; this allows it to be pulled out of the derivative term. As the derivative (time rate of change) of position is velocity and the derivative of velocity is acceleration. this equation can be simplified into several common forms,

with the most common form as F=ma, "force equals mass times acceleration", remember this equation is used independently in each direction, so F_x = m*a_x and F_y = m*a_y are separate equations. vectors can do a lot of cool things later, but for now, it's just properties in each direction.

For questions of what laws apply, the 2nd is applied for any non-static systems.

3. Every force that exists is applied on two bodies with the same magnitude and is colinear (on the same line) and is anti-normal (ew British word, it points the other way)

Basically, when drawing free body diagrams, every force can be depicted with two arrows pointing inward OR outward that are centered on the application point (or center of mass for gravity) and overlap at either the tips or tails if the drawings are close enough.

If you integrate force with respect to time the resulting value is called impulse, and is equal to the change in momentum. As two bodies have the same force at the same time for all time, and action between them generates the same impulse, just pointing in opposite directions. This means that for each action the change in momentum is also equal and opposite, so the total momentum of the system remains constant. This proves (not rigorously) the conservation of momentum as a FUNDAMENTAL PROPERTY OF THE UNIVERSE.

For questions of what laws apply, The 3rd applied is for whenever 2 bodies interact with forces. remember, they must be the same force, ie. NOT the normal force on the book and its weight, even if they would have the same magnitude.

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