KINEMATICS OF PARTICLES RELATIVE MOTION WITH RESPECT TO TRANSLATING AXES

Transcription

1 KINEMTICS OF PRTICLES RELTIVE MOTION WITH RESPECT TO TRNSLTING XES

2 In the previous articles, we have described particle motion using coordinates with respect to fixed reference axes. The displacements, velocities and accelerations so determined are termed absolute. It is not always possible or convenient, however, to use a fixed set of axes to describe or to measure motion.

3 In addition, there are many engineering problems for which the analysis of motion is simplified by using measurements made with respect to a moving reference system. These measurements, when combined with the absolute motion of the moving coordinate system, enable us to determine the absolute motion in question. This approach is called the relative motion analysis.

4 The motion of the moving coordinate system is specified with respect to a fixed coordinate system. In Newtonian mechanics, this fixed system is the primary inertial system, which is assumed to have no motion in space. For engineering purposes, the fixed system may be taken as any system whose absolute motion is negligible for the problem at hand.

5 For most Earthbound engineering problems, it is sufficiently precise to take for the fixed reference system a set of axes attached to the Earth, in which case we neglect the motion of the Earth.

6 For the motion of satellites around the Earth, a nonrotating coordinate system is chosen with its origin on the axis of rotation of the earth. For interplanetary travel, a nonrotating coordinate system fixed to the Sun would be used. Thus, the choice of the fixed system depends on the type of problem involved.

7 In this article, we will confine our attention to moving reference systems which translate but do not rotate. Motion measured in rotating systems will be discussed in rigid body kinematics, where this approach finds special but important application. We will also confine our attention here to relative motion analysis for plane motion.

8 Now let s consider two particles and which may have separate curvilinear motions in a given plane or in parallel planes; the positions of the particles at any time with respect to fixed OXY reference system are defined r r by and. Y O j y r r r / i x X Let s arbitrarily attach the origin of a set of translating (nonrotating) axes to particle and observe the motion of from our moving position on.

9 The position vector of as measured relative to the frame x-y is, where the subscript notation / means relative to or with respect to. The unit vectors along the x and y axes are and j r / = xi + yj, and x and y are the coordinates of measured in i Y O j y r r r / i x X the x -y frame. The absolute position of is defined by the vector r, measured from the origin of the fixed axes X -Y.

10 The position of is, therefore, determined by the vector r = r + r / Y O j y r r r / i x X

11 We now differentiate this vector equation = r + r once with respect to time to obtain r / velocities and twice to obtain accelerations. v = v + v ( r = r + r ) / / Here, the velocity which we observe to have from our position at attached to the moving axes x-y is r / v = / = xi + with respect to. yj. This term is the velocity of

12 cceleration is obtained as a = a + a, + / ( r = r + r ) ( v = v v ) / / Here, the acceleration which we observe to have from our nonrotating position on is r v = a = xi + / = / /. This term is the acceleration of with respect to. We note that the unit vectors and have zero i yj derivatives because their directions as well as their magnitudes remain unchanged. j

13 The velocity and acceleration equations state that the absolute velocity (or acceleration) equals the absolute velocity (or acceleration) of plus, vectorially, the velocity (or acceleration) of relative to. The relative term is the velocity (or acceleration) measurement which an observer attached to the moving coordinate system x-y would make.

14 We can express the relative motion terms in whatever coordinate system is convenient rectangular, normal and tangential or polar, and use their relevant expressions.

15 The selection of the moving point for attachment of the reference coordinate system is arbitrary. Point could have been used just as well for the attachment of the moving system, in which case the three corresponding relative-motion equations for position, velocity and acceleration are: Y O r v a r r j y = r = v = a r / + r / + v + a / i / x X

16 It is seen, therefore, that r / = r / v / = v / a / = a / In relative-motion analysis, it is important to realize that the acceleration of a particle as observed in a translating system x-y is the same as that observed in a fixed system X-Y if the moving system has a constant velocity. Y O r r j y r / i x X

17 1. Ferris wheel rotates at the instant of interest with an angular speed θ = 0.5 rad/s and is increasing its angular speed at the rate of 0.1 rad/s 2. ball is thrown from the ground to a person at. The ball arrives at the instant considered with a speed relative to the ground given as v = 2i 10 j. What are the velocity and acceleration of the ball relative to the person at, provided that this seat is not swinging? The radius of the wheel is 10 m. θ all R = 10 m

18 2. The aircraft with radar detection equipment is flying horizontally at an altitude of 12 km and is increasing its speed at the rate of 1.2 m/s each second. Its radar locks onto an aircraft flying in the same direction and in the same vertical plane at an altitude of 18 km. If has a speed of 1000 km/h at the instant when θ = 30, determine the values of r and θ at this same instant if has a constant speed of 1500 km/h.

19 3. Particles and both have a speed of 8 m/s along the directions indicated by arrows. moves in a curvilinear path defined by y 2 = x 3 and moves along a linear path defined by y = -x. If the velocity of is decreasing at a rate of 6 m/s each second and the velocity of is increasing at a rate of 5 m/s each second, determine the velocity and acceleration of with respect to for the instant represented.

20 4. Car is travelling along a circular path having a radius of 60 m with a constant speed of 54 km/h. t the instant passes from the position indicated, car is 30 m away from the junction and is increasing its speed of 72 km/h with 1.5 m/s 2. For the instant represented, determine the velocity and acceleration of with respect to an observer at. lso determine r, r, r, θ, θ, θ for this instant.

21 5. Small wheels attached to both ends of rod roll along smooth surfaces. t the instant shown, the wheel has a velocity of 1.5 m/s towards right and this speed is increasing at a rate of 0.5 m/s 2. Determine the absolute velocity of wheel ( v ), its relative velocity with respect to ( v / ), its absolute acceleration ( a ) and its relative acceleration with respect to ( ). a / 500 mm v mm