Mechanical Engineering Handbook P2

These equations are only to be used when the acceleration is known to be a constant. There are other expressions available depending on how a variable acceleration is given as a function of time, velocity, or displacement. | Mechanics of Solids 1-33 Useful Expressions Based on Acceleration Equations for nonconstant acceleration dv a dt v dv a dx v dv a dx v0 x0 Equations for constant acceleration projectile motion free fall v at v0 v2 2a x - x0 v0 dv f a dt v 0 1 2 . . . x at Kt x 2 0 0 These equations are only to be used when the acceleration is known to be a constant. There are other expressions available depending on how a variable acceleration is given as a function of time velocity or displacement. Scalar Relative Motion Equations The concept of relative motion can be used to determine the displacement velocity and acceleration between two particles that travel along the same line. Equation provides the mathematical basis for this method. These equations can also be used when analyzing two points on the same body that are not attached rigidly to each other Figure . FIGURE Relative motion of two particles along a straight line. xB A xB - XA vBA vB vA aBA aB - aA The notation B A represents the displacement velocity or acceleration of particle B as seen from particle A. Relative motion can be used to analyze many different degrees-of-freedom systems. A degree of freedom of a mechanical system is the number of independent coordinate systems needed to define the position of a particle. Vector Method The vector method facilitates the analysis of two- and three-dimensional problems. In general curvilinear motion occurs and is analyzed using a convenient coordinate system. Vector Notation in Rectangular Cartesian Coordinates Figure illustrates the vector method. 1-34 Section 1 FIGURE Vector method for a particle. The mathematical method is based on determining v and a as functions of the position vector r. Note that the time derivatives of unit vectors are zero when the xyz coordinate system is fixed. The scalar components x y x . can be determined from the appropriate scalar equations previously presented that only include the .

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