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Transcript
Free-Fall: An object that is dropped falls down due to the gravitational influence of Earth. If the object experiences no (or, at least, very little) air resistance, then it is in "free-fall". An object that is in free-fall increases its velocity by 9.8 m/s every second (this is equivalent to 32 feet/s every second). We therefore say that its acceleration is 9.8 m/s2. If an object is thus thrown downward initially with a velocity of, say, 5 m/s, then after 1 second its downward velocity is 5 + 9.8 = 14.8 m/s, after two seconds its downward velocity is 5 + 9.8 + 9.8 = 24.6 m/s, after three seconds its downward velocity is 5 + 9.8 + 9.8 + 9.8 = 34.4 m/s and so on. In equation form, we say that v = v0 - gt where v is the velocity of the object in free-fall after time t; v0 is the initial velocity of the object (positive if the object is thrown upwards, negative if the object is thrown downwards) and g represents the acceleration of the object due to gravity, 9.8 m/s2. It is possible for an object to be moving upwards and still be in free-fall. This will occur when the initial velocity of the object is positive, i.e., up. This also means that there is an acceleration acting downwards even when the object is moving upwards or even if the object has a velocity of zero. This again goes back to the situation of one dimensional motion and having a velocity of one sign and an acceleration of an opposite sign. The equation above is the relationship between velocity and acceleration when the acceleration is constant. We can also find a relationship between velocity and acceleration and displacement. We are aware that if the velocity is constant, then the change in displacement is d = vt where d is the change in displacement, v is the velocity and t is the time of travel. If, however, we started not at the origin but at some location d0 then our more general equation becomes d = d0 + vt. Now, though, suppose the velocity is changing, starting with an initial velocity of v0. It turns out that the relationship becomes d = d0 + v0t + ½at2. Note that the displacement depends upon the square of the time. We say that this formula is "quadratic". In many cases, we are able to say that the initial location of the object is 0, i.e., d0 = 0. Further, if an object starts from rest, then that means that v0 = 0. Thus, for an object starting at rest, the equation for distance traveled simplifies to d = ½at2. If the acceleration is known, as is the case for free-fall, then we can solve this equation in terms of t to get t = (2d/a)1/2; this can arise when we know the distance the object has traveled, but we wish to know for how long it has been traveling. In summary, the relevant equations are: xf = xi + vi t + ½ at2 vf = vi + at. A useful equation is vf2 = vi2 + 2a(xf – xi). (In these equations, the letter x essentially plays the same role as the letter d used in the discussion above.) Newton's First Law of Motion: Aristotle formulated the concept that the natural state of all bodies was to be at rest--after all, if I start anything moving, then that anything will eventually come to a rest. Unfortunately, Aristotle didn't have a good grasp of the effects of friction. Isaac Newton did. He realized that if all external forces were removed from an object, then the motion of that object would not change. This was Newton's first law of motion. In our study of motion, we need to start clarifying the concept of mass. In effect, we will find that mass is a measure of how hard it is to get an object moving. Applying a force to a light object will get the object into significant motion. Applying that same force to a heavy object will only seem to barely influence that object into motion. Also, in everyday-speak, we use the words "mass" and "weight" interchangeably. However, in physics-speak, they are two different quantities. "Weight" is dependent upon the gravity in which the object exists. For instance, since the Moon is lighter, the gravitational acceleration felt on the surface of the Moon is less than the gravitational acceleration felt on the surface of the Earth. Therefore, an object on the Moon weighs less than an object on the Earth. However, no matter where it is, that object has the same mass. Consider a baseball and a battleship in outer space where they would be weightless. If a given force is applied to the baseball, significant motion can be gained by the baseball. However, if that same force were applied to the battleship, the motion of the battleship would be only barely affected. That's because the mass of the battleship is so much more than the mass of the baseball. We can also say that the battleship has more inertia than the baseball. Newton's Second Law of Motion: You've likely noticed by this point in your life that not all motion acts in a straight line. From Newton's First Law, we know that if no forces acted on a moving object, it would move in a straight line. Therefore, if there is nonstraight motion, then there must be a force. Newton's Second Law gives us a chance to start quantifying the force and its effects on the moving object. Newton's Second Law states that a non-zero force will impart a non-zero acceleration to a mass. In fact, the acceleration that the mass gains is proportional to the force acting on it--if the force acting on the mass doubles, the acceleration of the mass doubles and if the force triples, the acceleration triples. The direction of the acceleration is also in the direction of the applied force. Also, the acceleration gained by the object is inversely proportional to the mass of the object, so if the mass of the object doubles, the acceleration is cut by one-half and if the mass of the object triples, the acceleration is cut in magnitude by one-third. In equation form, we say a = (Fnet/m) where the boldface notation indicates vector quantities and lets us know the acceleration is in the direction of the net force on the object. What do we mean exactly by the "net force" or Fnet? Suppose two forces act on the mass (honestly, this happens all the time--frequently, there are quite a few more than two different forces). We can represent these forces as arrows with lengths equal to the magnitude of each force, emanating from the object and each pointing in the direction of each force; in the most general situation, they won't be at right angles to each other. The net force is the vector sum of the two individual forces. We can show this graphically by considering each of the two vectors as the sides of a parallelogram with one vertex where the object is located. We can then complete the other two sides of the parallelogram and draw a diagonal from the vertex where the object is to the opposite vertex. This diagonal represents the net force acting on the object due to the two individual forces (see the diagram done in class for the graphical portion of this discussion). There is a mathema- tical way to come up with the exact value of this net force; this involves trigonometry and won't be discussed in this course, though.