The Description of Motion The frontiers of physics: Friction The disk of"dry ice shown in the multiple-exposure phot graph of figure 2.2 moves equal distances in equal times because friction between the puck and the smooth surface is negligible. When two surfaces rub together, the atoms that make up the two surfaces interact in ways that depend on the atomic composition of the substances, making them stick to each other. Although the general mechanism is well understood, the details of how friction appears are only now beginning to become clear. Recently, scientists at Georgia Institute of Technology used an atomic force micro- scope, which measures the forces between two objects sepa- rated by less than 10 nanometers, to examine the tip of a tiny nickel probe moving on a gold surface Friction seen at the atomic level between a nickel tip and a gold surface. (Scientific American. When the nickel tip was pulled back slightly after it had made contact with the surface, a connective"neck of atoms developed between the two surfaces, a sort of bridge at the atomic scale. After the tip was pulled far enough, the neck napped, leaving the tip covered with gold atoms. Why would gold atoms move over to the nickel tip instead of the other way around? Gold, it turns out, requires less energy to have one of its surface atoms removed than nickel. The researchers believe that these differences in energy account for the differences in friction between different substances
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SUPERSTRINGS AND OTHER THINGS for a few minutes until you have to slow down to make a turn or come to a stop at a traffic light. The speed at which you drive changes many times throughout your trip. We can obtain the average speed of the motion by dividing the total distance tra- veled by the time it took to cover that distance Lets consider a numerical example. A boy takes 15 minutes to ride his bicycle to his friend's house, which is 2km away. He talks to his friend for 20 minutes and then continues towards his grandparent's home, 4 additional km. He arrives there 25 minutes after leaving his friend,'s house ince the total distance traveled is 6 km(2 km to the friends house and 4 to the grandparents home)and it took the boy a total of 60 minutes to get there, the average speed is d 6km U 6 km/h min Notice that we have included the time the boy spends at his friends house in our calculation of the total time taken for the trip Instantaneous speed Average speed is useful information. When we are traveling, we can calculate the average speed for a section of the trip and use it to estimate how long it will take us to complete the trip, provided we continue driving under similar conditions. However, in some cases, we might be interested in obtaining more information. It would take you about 8 hours to drive from Washington, DC to Charlotte, North Carolina, a distance of 400 miles. Although the average speed in this case is 50 mph, you know that at times ou would drive at a higher speed, whereas heavy traffic or lower speed limits through certain parts would force you to drive at a lower speed. Knowing that you can average 50 mph for this trip does not provide information about how fast you actually traveled or whether you stopped at all along the way The instantaneous speed, when we can obtain it, will give us infor mation about the detail of the trip. Instantaneous speed is the speed given by a cars speedometer, the speed at a given instant If your car speedometer fails before you complete a 60-mile trip that usually takes you one hour, you cannot be sure that 30
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The Description of Motion you did not break the 65 mph speed limit at any time during your trip even if it still took you one hour to arrive at your destination. However, if you time yourself between successive mile posts and it takes you about one minute to travel one mile, you know that you are maintaining a speed close to 60 mph. To measure your speed more accurately you would need to reduce the time inter vals to one second, perhaps-during which you would travel only about 90 feet-or, with precision equipment, even to 1/10 second Yet, this very small interval of time would still not give you the instantaneous speed. You would need to reduce that interval to an instant Modern instrumentation allows measurements of the speed of an object at intervals small enough to provide us with excellent approximations to the instantaneous speed. Mathematically, it is possible to obtain the exact value of the instantaneous speed by the use of calculus, a mathematical technique developed over 300 years ago by Isaac Newton. Velocity: Speed and direction In some cases the direction in which we are moving is also impor tant information a pilot needs to know how fast the wind is blow- ing and in what direction. The pilot needs to know the wind velocity. Velocity gives the speed and the direction of motion. The wind velocity of a 50-mph wind blowing east would push off course a small airplane heading north at 80 mph. If the ilot did not correct the airplane's heading it would end up flying in a northeasterly direction Velocity, and other quantities that require a magnitude and a direction, are called vector quantities. In the next section we shall study some properties of these new quantities. Quantities that do not require a direction are said to be scalar quantities. Speed is a scalar quantity Vectors As we have said, vector quantities are those that possess both magnitude and a direction. We represent vectors as arrows
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SUPERSTRINGS AND OTHER THINGS with length proportional to the magnitude and with a direction indicating the direction of the vector quantity. We use bold face letters,(v, V)to completely represent vectors and standard letters (U, V)to indicate their magnitudes. To illustrate some properties of vectors, let,'s consider the following situation. A man wants to buy a paperback he has heard about recently. He walks 3 km east to his friends house and then both walk together to the nearest drugstore, 4 km from his friends house and in the same direction(figure 2.3(a) 4 km 3 km 7 km Figure 2.3.(a)A man walks 3 km to his friends house and then 4 km to the drugstore. The man is 7 km away from home. If instead he walks 4km from the friends house to the library, the man is only 5km from home (b) The two individual trips of 3 and 4 km are equal to a single trip of 7km.(c) Walking east 3 km (labeled with vector a) and then north 4 km(vector b) is equivalent to walking across in the direction show 32
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Obviously, the total distance traveled by the man is 7km. Since the two trips take place in the same direction, the man find himself 7km away from home. Graphically, we can illustrate this situation as in figure 2.3(b) Let's suppose now that the man wants to borrow the book from the local library instead of buying it at the drugstore (figure 2.3(a)). If the library is also 4 km from his friend's house but north instead of east, as illustrated in figure 2.3(b), the man is now only 5km from home. Using Pythagoras's Theorem we get a displacement of v(4*+3)km=5km. If he had wanted, he could have cut across the field straight from his home to the library, walking only the 5km. If vector a represents the 3-km walk east, and b the 4-km walk north, vector c represents the straight walk across the field from the house to the library. Vector c is equivalent to the two vectors a and b together In other words walking east for 3 km to the friends house and then north for 4 km to the library is the same as walking 5 km across the field from the house to the library. We call vector c the resultant of vectors a and b and the process addition of vectors An alternative method, the so-called parallelogram method for the addition of vectors, consists of placing the vectors to be added tail-to-tail"instead of"head-to-tail, keeping their orientations fixed. The resultant is obtained by completing the diagonal of the parallelogram. In figure 2.4 we illustrate the addition of a Figure 2.4. Addition of two vectors using the parallelogram method
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