As illustrated in the accompanying figure, suppose that the equations of motion of a particle moving along an elliptic path are x = a cos w t , y = b sin w t . (a) Show that the acceleration is directed toward the origin. (b) Show that the magnitude of the acceleration is proportional to the distance from the particle to the origin.
As illustrated in the accompanying figure, suppose that the equations of motion of a particle moving along an elliptic path are x = a cos w t , y = b sin w t . (a) Show that the acceleration is directed toward the origin. (b) Show that the magnitude of the acceleration is proportional to the distance from the particle to the origin.
As illustrated in the accompanying figure, suppose that the equations of motion of a particle moving along an elliptic path are
x
=
a
cos
w
t
,
y
=
b
sin
w
t
.
(a) Show that the acceleration is directed toward the origin.
(b) Show that the magnitude of the acceleration is proportional to the distance from the particle to the origin.
The motion of a point on the circumference of a rolling wheel of radius 4 feet is described by the vector
function
r(t) = 4(12t - sin(12t))i + 4(1 − cos(12t))j
Find the velocity vector of the point.
v(t) =
Find the acceleration vector of the point.
a(t) =
Find the speed of the point.
s(t) =
=
The motion of a vibrating particle is defined by the position vector r = (4 sin nt)i – (cos 2ntj, where ris expressed in inches
and tin seconds. (a) Determine the velocity and acceleration when t= 1 s. (b) Show that the path of the particle is
parabolic.
Fig. P11.91
1 in.
11 in.
A particle moves in space along a path whose parametric equations are given by x1 =
b sin(wt), x2 = b cos(wt), x3 = c where b and c are constants.
a. Find its position at vector 7, velocity v, and acceleration å at any time t.
b. Show that the particle traverses its path with constant speed and that its
distance from the origin remains constant.
c. Show that the acceleration is perpendicular to the velocity and the x3-axis.
d. Determine the trajectory of the particle, that is, the path described in terms
of spatial coordinates only. Sketch the trajectory.
1.
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