Transcript Slide 1
Plane curvilinear motion is the motion of a particle along a curved path which lies in a single plane. Before the description of plane curvilinear motion in any specific set of coordinates, we will use vector analysis to describe the motion, since the results will be independent of any particular coordinate system. At time t the particle isat position A, which is located by the position vector r measured from the fixed origin O. Both the magnitude and direction of r are known at time t. At time t+Dt, the particle is at A' , r D r located by the position vector . V Path of particle r Dr r O t+Dt A' A' Dr V Ds A t DV AV V a A The displacement of the particle during Dt is the vector D r which represents the vector change of position and is independent of the choice of origin. If another point was selected as the origin the position vectors would have changed but D r would remain the same. V Path of particle r Dr r t+Dt A' A' Dr V Ds A t DV AV V a A O The distance actually travelled by the particle as it moves along the path from A to A' is the scalar length Ds measured along the path. It is important to distinguish between Ds and D r . Velocity The average velocity of the particle between A and A' is defined as Dr Dr vav Dt which is that is a vector whose direction of D r . The magnitude of vav is Dr . Dt The average speed of the particle between A and A' is Ds Dt Clearly, the magnitude of the average Ds Dr velocity Dt and the speed Dt approach one another as the interval Dt decreases and A and A' become closer together. The instantaneous velocity v of the particle is defined as the limiting value of the average velocity as the time Dt approaches zero. Dr dr v lim r dt Dt 0 Dt Dr We observe that the direction of v approaches that of the tangent to the path as Dt approaches zero and, thus, the velocity is always a vector tangent to the path. The magnitude of v is called the speed and is the scalar ds v v s dt The change in velocities, which are tangent to the path and are v at A and v at A‘ during time Dt is a vector Dv . v Dv v D v Here indicates both change in magnitude and direction of v . Therefore, when the differential of a vector is to be taken, the changes both in magnitude and direction must be taken into account. Acceleration The average acceleration of the particle A and A' is defined as aav Dv Dt which is a vector whose direction is that D v of Dv . Its magnitude is Dt The instantaneous acceleration of the particle is defined as the limiting value of the average acceleration as the time interval approaches zero. Dv dv a lim vr dt Dt 0 Dt As Dt becomes smaller and approaches zero, the direction of Dv approaches dv . The acceleration includes the effects of both the changes in magnitude and direction of v . In general, the direction of the acceleration of a particle in curvilinear motion is neither tangent to the path nor normal to the path. If the acceleration was divided into two components one tangent and the other normal to the path, it would be seen that the normal component would always be directed towards the center of curvature. If velocity vectors are plotted from some arbitrary point C, a curve, called the hodograph, is formed. Acceleration vectors are tangent to the hodograph. Three different coordinate systems are commonly used in describing the vector relationships for plane curvilinear motion of a particle. These are: • Rectangular (Cartesian) Coordinates (Kartezyen Koordinatlar) • Normal and Tangential Coordinates (Doğal veya Normal-Teğetsel Koordinatlar) • Polar Coordinates (Polar Koordinatlar) The selection of the appropriate reference system is a prerequisite for the solution of a problem. This selection is carried out by considering the description of the problem and the manner the data are given. Cartesian Coordinate system is useful for describing motions where the x- and y-components of acceleration are independently generated or determined. Position, velocity and acceleration vectors of the curvilinear motion is indicated by their x and y components. y j yj O v a vx ax Path of particle vy A r xi q ay a i A x Let us assume that at time t the particle is at point A. With the aid of the unit vectors i and ,j we can write the position, velocity and acceleration vectors in terms of xand y-components. r xi yj v v x i v y j xi yj a a x i a y j v x i v y j xi yj As we differentiate with respect to time, we observe that the time derivatives of the unit vectors are zero because their magnitudes and directions remain constant. The magnitudes of the components of v and a are: v x x a x v x x v y y a y v y y In the figure it is seen that the direction of ax is in –x direction. Therefore when writing in vector form a “-” sign must be added in front of ax. The direction of the velocity is always tangent to the path. No such thing can be said for acceleration. v 2 vx 2 v y 2 v vx 2 v y 2 tanq vy vx a 2 ax 2 a y 2 a ax 2 a y 2 tana ay ax If the coordinates x and y are known independently as functions of time, x=f1(t) and y=f 2(t), then for any value of the time we can obtain r .Similarly, we combine their first derivatives x and y to obtain v and their second derivatives x and y to obtain a . Inversely, if ax and ay are known, then we must take integrals in order to obtain the components of velocity and position. If time t is removed between x and y, the equation of the path can be obtained as y=f(x). Projectile Motion (Eğik Atış Hareketi) An important application of two-dimensional kinematic theory is the problem of projectile motion. For a first treatment, we neglect aerodynamic drag and the curvature and rotation of the earth, and we assume that the altitude change is small enough so that the acceleration due to the gravity can be considered constant. With these assumptions, rectangular coordinates are useful to employ for projectile motion. Acceleration components; ax=0 ay= -g Apex; vy=0 y vy voy= vosinq vo q vox= vocosq vx v vx vx g v'y v' x If motion is examined separately in horizontal and vertical directions, Horizontal ax 0 v x constant Vertical ay -g v y v0 y - gt 1 2 gt 2 v x v0 x y y 0 v0 y t - x x 0 v0 x t v y 2 v0 y 2 - 2 g ( y - y 0 ) We can see that the x- and y-motions are independent of each other. Elimination of the time t between x- and y-displacement equations shows the path to be parabolic. 1. The particle P moves along the curved slot, a portion of which is shown. Its distance in meters measured along the slot is given by s=t2/4, where t is in seconds. The particle is at A when t = 2.00 s and at B when t =2.20 s. Determine the magnitude aav of the average acceleration of P between A and B. Also express the acceleration as a vector aav using unit vectors i and j . 2. With what minimum horizontal velocity u can a boy throw a rock at A and have it just clear the obstruction at B? 3. For a certain interval of motion, the pin P is forced to move in the fixed parabolic slot by the vertical slotted guide, which moves in the x direction at the constant rate of 40 mm/s. All measurements are in mm and s. Calculate the magnitudes of v and a of pin P when x = 60 mm. 4. A projectile is fired with a velocity u at right angles to the slope, which is inclined at an angle q with the horizontal. Derive an expression for the distance R to the point of impact. 5. Pins A and B must always remain in the vertical slot of yoke C, which moves to the right at a constant speed of 6 cm/s. Furthermore, the pins cannot leave the elliptic slot. What is the speed at which the pins approach each other when the yoke slot is at x = 50 cm? What is the rate of change of speed toward each other when the yoke slot is again at x = 50 cm? yoke 60 cm 100 cm 6 cm/s x 6. A 7 m long conveyor band makes an angle a with the horizontal surface. Sand, thrown at point B freely falls to the point C on the surface. If the band is moving with a constant velocity v0 = 3 m/s, calculate the maximum distance d between A and C and also find a. 7m B A C d