Transcript Slide 1
KINEMATICS OF PARTICLES RELATIVE MOTION WITH RESPECT TO TRANSLATING AXES 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 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”. In this article, we will confine our attention to moving reference systems which translate but do not rotate. Now let’s consider two particles A and B 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 Translating axis system are defined by rA and rB . Fixed axis Let’s attach the origin of a set of translating (nonrotating) axes to particle B and observe the motion of A from our moving position on B. The position vector of A as measured relative to the frame x-y is rA / B xi yj , where the subscript notation “A/B” means “A relative to B” or “A with respect to B”. Translating axes Fixed axes The position of A is, therefore, determined by the vector rA rB rA / B Translating axes Fixed axes We now differentiate this vector equation rA rB rA / B once with respect to time to obtain velocities and twice to obtain accelerations. v A vB v A / B rA rB rA / B Here, the velocity which we observe A to have from our position at B attached to the moving axes x-y is rA/ B vA/ B xi yj This term is the velocity of A with respect to B. Acceleration is obtained as rA rB rA / B , a A aB a A / B vA vB vA / B Here, the acceleration which we observe A to have from our nonrotating position on B is . rA/ B vA/ B aA/ B xi yj This term is the acceleration of A with respect to B. We note that the unit vectors i and j have zero derivatives because their directions as well as their magnitudes remain unchanged. We can express the relative motion terms in whatever coordinate system is convenient – rectangular, normal and tangential or polar, and use their relevant expressions. 1. The car A has a forward speed of 18 km/h and is accelerating at 3 m/s2. Determine the velocity and acceleration of the car relative to observer B, who rides in a nonrotating chair on the Ferris wheel. The angular rate = 3 rev/min of the Ferris wheel is constant. 4. Car A is traveling along a circular curve of 60 m radius at a constant 30 speed of 54 km/h. When A passes the B o 60 m position shown, car B is 30 m from the intersection, traveling with a speed of 72 km/h and accelerating at the rate of 1.5 m/s2. Determine the velocity and acceleration which A appears to have when observed by an occupant of B at this instant. Also determine r, q, instant. r , q , r and q for this r 30 m q A