Begin2.DVI
If r represents the length and direction of a line drawn to the center of mass of a body, then dr dt =v represents the instan ...
Example 6-19. A cannon ball of mass mis fired from a cannon with an initial velocity v 0 inclined at an angle θwith the horizont ...
where c 1 , c 2 , c 3 , c 4 are constants of integration. The solution satisfying the initial conditions can be expressed as y=y ...
and so it is placed at the end of the position vector, as illustrated in the figure 6-13 to show that the velocity is tangent to ...
where the last simplification was obtained using the vector identity given by equation (6.32) and the result ω·r = 0. The above ...
The physical interpretation applied to the acceleration vector is as follows. Ob- serve that the vectors ˆer= cos θeˆ 1 + sin θˆ ...
The velocity vector is in the direction of the tangent to the curve and the com- ponent of the velocity along the direction 0 P ...
Equating like components produces the result that r dθ ds = sin ψ and dr ds = cos ψ The derivative of the position vector r =re ...
Comparing these last two equations it is found that the time rate of change of angular momentum is expressible in terms of the f ...
through the angle φis given by s=aφ. The magnitude of the linear speed v, of the point P, is given by v=ds dt =adφ dt =aω =|v | ...
It therefore remains to show that v 1 =v. The geometry of figure 6-14, provides an aid in demonstrating that the vectors r 1 ...
origin, which sweeps out the curve as the parameter xvaries. In figure 6-15, the position vector r is illustrated. This positio ...
Figure 6-15. Tangent line and normal line to plane curve change with position. Recall from our earlier study of calculus that th ...
By using the results tan θ=dy dx and ds^2 =dx^2 +dy^2 ,one can calculate the derivatives dθ dx = d^2 y dx^2 1 + (dy dx ) 2 and d ...
The reciprocal of the curvature κis called the radius of curvature ρ. Note that straight lines have a constant angle θbetween a ...
From this result the center of curvature is found to have the position vector C(x) = r (x) + αN(x) = xˆe 1 +f(x)ˆe 2 +1 + [f ...
so that dη dξ =−ξ−h η−k =f′(x) and d (^2) η dξ^2 =− 1 + (dη dξ ) 2 η−k =f′′(x) This shows that the first and second derivatives ...
having a depth h as illustrated. Construct a set of x, y axes with y= 0 represent- ing the bottom of the channel and y=h represe ...
If these direction are the same, then by equating like components one must have dx =ky and dy =kx or dx y = dy x =k (6 .79) The ...
Figure 6-17. Contour plots of selected two-dimensional scalar functions. A vector field is a one–to–one correspondence between p ...
«
1
2
3
4
5
6
7
8
9
10
»
Free download pdf