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.
An object is spinning at a constant speed on the end of a string, according to the position vector r(t) = a cos ωti + a sin ωtj. (a) When the angular speed ω is doubled, how is the centripetal component of acceleration changed? (b) When the angular speed is unchanged but the length of the string is halved, how is the centripetal component of acceleration changed?
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) =
=
A particle moves so that its position vector at time t is given by
r = e cos ti + e sin tj.
Show that at any time t,
(a) its velocity v is inclined to the vector r at a constant angle 37/4
radians.
(b) its acceleration vector is at right angles to the vector r.
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