Math 1320 Lab 9
Name:
1. Consider a particle traveling along the path he−t cos(t), e−t sin(t), e−t i for 0 ≤ t < ∞.
(a) Sketch or use a computer to visualize the trajectory of the particle. For instance,
you could try the Wolfram Alpha input:
ParametricPlot3D[{x(t), y(t), z(t)}, {t, 0, 5}]
with the appropriate functions. You do not need to write anything for this part,
but I encourage you to visualize the curve.
(b) At time T , what is the distance that the particle has traveled? As T → ∞, what is
the total distance that the particle travels?
2. What is the curvature of f (x) =
1
at x = 0?
1 + x2
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3. What is the curvature of the ellipse
curvature constant?
x 2
a
+
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y 2
b
= 1? Under what condition is the
4. A particle P moves with constant angular speed ω around a circle whose center is at
the origin and whose radius is R. The particle is said to be in uniform circular motion.
Assume that the motion is counterclockwise and that the particle is at the point (R, 0)
when t = 0. The position vector at time t ≥ 0 is ~r(t) = hR cos ωt, R sin ωti.
(a) Find the velocity vector ~v (t) and show that ~v · ~r = 0. Conclude that ~v is tangent to
the circle and points in the direction of the motion.
(b) Show that the speed |~v | of the particle is the constant ωR.
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(c) Find the acceleration vector ~a. Show that it is proportional to ~r. Then show that it
points towards the origin. An acceleration with this property is called a centripetal
acceleration. Show that the magnitude of the acceleration vector is |~a| = Rω 2 .
(d) Suppose that the particle has mass m. Show that the magnitude of the force F~
that is required to produce this motion (called a centripetal force), is
|F~ | =
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m|~v |2
.
R