KINEMATICS OF PARTICLES RESPECT TO TRANSLATING AXES

Transcription

1 KINEMATICS OF PARTICLES RELATIVE MOTION WITH RESPECT TO TRANSLATING AXES

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.

3 It is not always possible or convenient to use a fixed set of axes to describe or to measure motion. 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 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.

5 Now let s consider two particles A 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 by and. r v A r v Translating axis Fixed axis Let s attach the origin of a set of translating (nonrotating) axes to particle and observe the motion of A from our moving position on.

6 The position vector of A as measured r A / v yj relative to the frame x-y is, where the subscript notation A/ means A relative to or A with respect to. v v = xi + Translating axes Fixed axes

7 The position of A is, therefore, determined by the vector v v v r = r + r A A/ Fixed axes Translating axes

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

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

10 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 A relative to. The relative term is the velocity (or acceleration) measurement which an observer attached to the moving coordinate system x-y would make. We can express the relative motion terms in whatever coordinate system is convenient rectangular, normal and tangential or polar, and use their relevant expressions.

11 1. The car A has a forward speed of 18 km/h and is accelerating at 3 m/s 2. Determine the velocity and acceleration of the car relative to observer, who rides in a nonrotating chair on the Ferris wheel. The angular rate Ω = 3 rev/min of the Ferris wheel is constant.

12 2. A batter hits the baseball A with an initial velocity of v 0 =30 m/s directly toward fielder at an angle of 30 to the horizontal; the initial position of the ball is 0.9 m above ground level. Fielder requires ¼ s to judge where the ball should be caught and begins moving to that position with constant speed. ecause of great experience, fielder chooses his running speed so that he arrives at the catch position simultaneously with the ball. The catch position is the field location at which the ball altitude is 2.1 m. Determine the velocity of the ball relative to the fielder at the instant the catch is made.

13 3. Airplane A is flying horizontally with a constant speed of 200 km/h and is towing the glider, which is gaining altitude. If the tow cable has a length r = 60 m and θ is increasing at the constant rate of 5 degrees per second, determine the magnitudes of the velocity v and acceleration a v of the glider for the instant when θ = 15.

14 4. Car A is traveling along a circular curve of 60 m radius at a constant speed of 54 km/h. When A passes the position shown, car is 30 m from the intersection, traveling with a 30 m 30 o θ 60 m r A speed of 72 km/h and accelerating at the rate of 1.5 m/s 2. Determine the velocity and acceleration which A appears to have when observed by an occupant of at this instant. Also r& & θ & r& determine r, θ,,, and & θ for this instant.