Principios de mezclado

While the vector addition cussed in Chapter 3 involves dis-placement vectors, vector addition can be applied to any type of vector quantity. Figure 4.4, for example, shows the addition of velocity vectors using the graphi-cal approach.

where x, y, and r change with time as the particle moves while the unit vectors iˆ and jˆ remain constant. If the position vector is known, the velocity of the particle can be ob-tained from Equations 4.3 and 4.6, which give

(4.7) Because a is assumed constant, its components a_xand a_yalso are constants. Therefore, we can apply the equations of kinematics to the x and y components of the velocity vec-tor. Substituting, from Equation 2.9, v_xf"v_xi&a_xt and v_yf"v_yi&a_yt into Equation 4.7 to determine the final velocity at any time t, we obtain

(4.8) This result states that the velocity of a particle at some time t equals the vector sum of its initial velocity v_iand the additional velocity at acquired at time t as a result of con-stant acceleration. It is the vector version of Equation 2.9.

Similarly, from Equation 2.12 we know that the x and y coordinates of a particle moving with constant acceleration are

Substituting these expressions into Equation 4.6 (and labeling the final position vectorr_f) gives

(4.9) which is the vector version of Equation 2.12. This equation tells us that the position vector r_f is the vector sum of the original position r_i, a displacement v_it arising from the initial velocity of the particle and a displacement at²resulting from the constant acceleration of the particle.

Graphical representations of Equations 4.8 and 4.9 are shown in Figure 4.5. Note from Figure 4.5a that v_fis generally not along the direction of either v_ior a because the relationship between these quantities is a vector expression. For the same reason,

12

r_f "r_i&v_it &¹₂at²

" (x_iiˆ & y_i jˆ) & (v_xiiˆ & v_yi jˆ)t &¹₂(a_xiˆ & a_y jˆ)t² r_f "(x_i&v_xit &¹₂a_xt²)iˆ & (y_i&v_yit &¹₂a_yt²)jˆ

x_f"x_i&v_xit &¹₂a_xt²

^{yf"yi&vyit &1}2a_yt² v_f "v_i&at

" (v_xiiˆ & v_yijˆ)& (a_xiˆ & a_yjˆ)t v_f "(v_xi&a_xt)iˆ & (v_yi&a_yt)jˆ v " dr

dt " dx dt iˆ & dy

dt jˆ " v_x iˆ & v_yjˆ

y

x a_yt

v_yf

v_yi

v_f

v_i at

v_xi a_xt

v_xf (a)

y

x y_f

y_i

r_f

v_it

v_xit x_f (b) a_yt²

12

v_yit

r_i

at²

12

a_xt²

12

x_i

Active Figure 4.5 Vector representations and components of (a) the velocity and (b) the posi-tion of a particle moving with a constant acceleraposi-tion a.

Velocity vector as a function of time

Position vector as a function of time

At the Active Figures link at http://www.pse6.com, you can investigate the effect of different initial positions and velocities on the final position and velocity (for constant acceleration).

Example 4.1 Motion in a Plane

A particle starts from the origin at t " 0 with an initial veloc-ity having an x component of 20 m/s and a y component of # 15 m/s. The particle moves in the xy plane with an x component of acceleration only, given by a_x"4.0 m/s². (A) Determine the components of the velocity vector at any time and the total velocity vector at any time.

Solution After carefully reading the problem, we conceptu-alize what is happening to the particle. The components of the initial velocity tell us that the particle starts by moving toward the right and downward. The x component of veloc-ity starts at 20 m/s and increases by 4.0 m/s every second.

The y component of velocity never changes from its initial value of # 15 m/s. We sketch a rough motion diagram of the situation in Figure 4.6. Because the particle is accelerat-ing in the & x direction, its velocity component in this direc-tion will increase, so that the path will curve as shown in the diagram. Note that the spacing between successive images increases as time goes on because the speed is increasing.

The placement of the acceleration and velocity vectors in Figure 4.6 helps us to further conceptualize the situation.

Because the acceleration is constant, we categorize this problem as one involving a particle moving in two dimen-sions with constant acceleration. To analyze such a problem, we use the equations developed in this section. To begin the mathematical analysis, we set vxi"20 m/s, vyi" #15 m/s, a_x"4.0 m/s², and a_y"0.

Equations 4.8a give

(1) vxf"vxi&axt " (20 & 4.0t) m/s

(2) v_yf"v_yi&a_yt " #15 m/s & 0 " #15 m/s Therefore

We could also obtain this result using Equation 4.8 di-rectly, noting that a " 4.0iˆ m/s²and vi"[20iˆ # 15jˆ] m/s.

To finalize this part, notice that the x component of velocity increases in time while the y component remains constant;

this is consistent with what we predicted.

(B) Calculate the velocity and speed of the particle at t " 5.0 s.

Solution With t " 5.0 s, the result from part (A) gives

This result tells us that at t " 5.0 s, vxf"40 m/s and v_yf" #15 m/s. Knowing these two components for this two-dimensional motion, we can find both the direction and the magnitude of the velocity vector. To determine the angle ' that v makes with the x axis at t " 5.0 s, we use the fact that tan ' " v_yf/v_xf:

"

where the negative sign indicates an angle of 21° below the positive x axis. The speed is the magnitude of v_f:

"

To finalize this part, we notice that if we calculate vifrom the x and y components of v_i, we find that v_f ( v_i. Is this con-sistent with our prediction?

43 m/s

v_f"" v_f" "

√

^vxf²&v_yf²"

√

^{(40)2&(#15)2 m/s}

#21)

(3)

^{' "tan#1}

#

^vvyfxf

$

" tan^#1

#

^{#15 m/s}40 m/s

$

(40iˆ # 15jˆ) m/s v_f"%(20 & 4.0(5.0))iˆ # 15jˆ&m/s "

%(20 & 4.0t)iˆ # 15jˆ& m/s v_f"v_xiiˆ & v_yijˆ "

from Figure 4.5b we see that r_fis generally not along the direction of v_ior a. Finally, note that v_fand r_fare generally not in the same direction.

Because Equations 4.8 and 4.9 are vector expressions, we may write them in compo-nent form:

(4.8a)

(4.9a) These components are illustrated in Figure 4.5. The component form of the equations for v_fand r_fshow us that two-dimensional motion at constant acceleration is equivalent to two independent motions—one in the x direction and one in the y direction—having constant accelerations a_xand a_y.

r_f"r_i&v_it &¹₂at²

'

^{xyff""yxii&&vvyixit &t &1}2¹²a^a_y^xt^t22 v_f"v_i&at

'

^{vvxfyf""vvyixi&&aayxtt}

x y

Figure 4.6 (Example 4.1) Motion diagram for the particle.

(C) Determine the x and y coordinates of the particle at any time t and the position vector at this time.

Solution Because x_i"y_i"0 at t " 0, Equation 4.9a gives

Therefore, the position vector at any time t is

(Alternatively, we could obtain r_fby applying Equation 4.9 directly, with vf"(20iˆ # 15jˆ) m/s and a " 4.0iˆ m/s². Try it!) Thus, for example, at t " 5.0 s, x " 150 m, y " # 75 m, and rf"(150iˆ # 75jˆ) m. The magnitude of the displace-ment of the particle from the origin at t " 5.0 s is the mag-nitude of rf at this time:

Note that this is not the distance that the particle travels in this time! Can you determine this distance from the available data?

r_f"" r_f" "

√

^{(150)2&(#75)2}m " 170m

%(20t & 2.0t²)iˆ # 15t jˆ& m (4)

^{rf"xfiˆ & yfjˆ "}

(#15t) m y_f"v_yit "

(20t & 2.0t²) m x_f"v_xit &¹₂a_xt²"

To finalize this problem, let us consider a limiting case for very large values of t in the following What If?

What If? What if we wait a very long time and then observe the motion of the particle? How would we describe the mo-tion of the particle for large values of the time?

Answer Looking at Figure 4.6, we see the path of the parti-cle curving toward the x axis. There is no reason to assume that this tendency will change, so this suggests that the path will become more and more parallel to the x axis as time grows large. Mathematically, let us consider Equations (1) and (2). These show that the y component of the velocity re-mains constant while the x component grows linearly with t.

Thus, when t is very large, the x component of the velocity will be much larger than the y component, suggesting that the velocity vector becomes more and more parallel to the x axis.

Equation (3) gives the angle that the velocity vector makes with the x axis. Notice that ' : 0 as the denomina-tor (v_xf) becomes much larger than the numerator (v_yf).

Despite the fact that the velocity vector becomes more and more parallel to the x axis, the particle does not approach a limiting value of y. Equation (4) shows that both x_f and y_f con-tinue to grow with time, although x_f grows much faster.

In document COMPORTAMIENTO DE LA MATRIZ POLIMERICA PVC/ABS CON EL SISTEMA NANOPARTICULADO ZrO2/SiO2 (página 34-38)