Math 227 Problem Set II
Due Wednesday, January 20
1. Find the curvature of the plane curve y = ex .
2. Find the minimum and maximum values for the curvature of the ellipse x(t) = a cos t, y(t) = b sin t.
Here a > b > 0.
3. Let r = f (θ) be the equation of a plane curve in polar coordinates. Find the curvature of this curve at
a general point θ.
4. Find the curvature of the cardiod r = a(1 − cos θ).
5. Find the unit tangent, unit normal and binormal vectors and the curvature and torsion of the curve
r(t) = tı̂ı +
t2
2
̂ +
t3
3
k̂
6. (a) Show that if a curve has curvature κ(s) = 0 for all s, then the curve is a straight line.
(b) Show that if a curve has torsion τ (s) = 0 for all s, then the curve lies in a plane.
(c) Show that if a curve has curvature κ(s) = κ0 , a strictly positive constant, and torsion τ (s) = 0 for
all s, then the curve is a circle.
7. Solve the initial value problem
dr
dt
= k̂ × r
r(0) = ı̂ı + k̂
Describe the curve r = r(t).
8. Find the curvature κ as a function of arclength s (measured from (0, 0)) for the curve
x(θ) =
Z
0
θ
cos
2
1
2 πt
dt
y(θ) =
Z
θ
0
sin
2
1
2 πt
dt
9. A frictionless roller–coaster track has the form of one turn of the circular helix with parametrization
(a cos θ, a sin θ, bθ). A car leaves the point where θ = 2π with zero velocity and moves under gravity to
the point where θ = 0. By Newton’s law of motion, the position r(t) of the car at time t obeys
mr′′ (t) = N r(t) − mg k̂
Here m is the mass of the car, g is a constant, −mg k̂ is the force due to gravity and N r(t) is the
force that the roller–coaster track applies to the car to keep the car on the track. Since the track is
frictionless, N r(t) is always perpendicular to v(t) = dr
dt (t).
1
2
(a) Prove that E(t) = 2 m|v(t)| +mgr(t)· k̂ is a constant, independent of t. (This is called “conservation
of energy”.)
(b) Prove that the speed |v| at the point θ obeys |v|2 = 2gb(2π − θ).
(c) Find the time it takes to reach θ = 0.