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Unformatted text preview: UNIT 9 VARIABLE ACCELERATION, ARC LENGTH AND TANGENT VECTOR INTRO: In this Unit we will introduce three different kinds of problems. First we will discuss variable acceleration problems, which arise in problems with time-dependent forces. These will be solved by following the same three steps taken to solve projectile problems. Next, we will learn how to measure the distance along a curve, ie., arc length . Finally, we will begin the study of coordinate systems aligned with trajectories. 1. Variable Acceleration Problems with time-dependent forces, leading to time-dependent accelerations, can be treated in the same manner as the projectile problems with constant (gravitational) force treated in the previous Unit, in the sense that the initial conditions can first be written down, the acceleration can then be integrated twice to find the velocity and the position, and finally the particular question can be answered. We will illustrate this with three examples. If a particle with mass m and charge q is moving in an electric field, then the force on the particle will be ~ F = q ~ E . If the electric field has units of newtons/coulomb and the charge is in coulombs, then obviously the force will have units of newtons, although in the quiz problems we may choose to omit indicating units explicitly. If the electric field varies with time, then the force will be time dependent, and therefore so will the acceleration: ~a = 1 m ~ F = q m ~ E Such motion will lead to trajectory problems with time dependent acceleration. Note that in physical problems, the electric force is so much stronger than gravitational forces that gravitation can be ignored to a very great degree of accuracy. The unit of charge in electrical engineering is nearly always the coulomb. In physics the unit of charge is usually also the coulomb, although it is not unusual for the unit of charge to be taken such that the ratio of the charge of an electron to the mass of an electron will be -1. Example 1 : A particle with charge 3 and mass 8 is traveling in the time dependent electric field ~ E = 4 ˆ i +10sin t ˆ j , where t is the time in seconds. If the particle is instantaneously at rest at position (1 , 4 , 1), where will it be 10 seconds later? First write the intial conditions. ~ r (0) = < 1 , 4 , 1 > ~v (0) = < , , > Next write the acceleration and integrate to find velocity and position. Because the acceleration is time-dependent, we will have to set the integration constants with more care than we did in projectile problems, since the integrals of the acceleration may give more complicated functions VECTOR GEOMETRY 105 GREENBERG than the simple polynomials in t which arose in projectile problems with constant acceleration....
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