In this problem, we will find expressions for velocity and acceleration in cylindrical coordinates. We begin with the expression
x(t) = x(t)i + y(t)j + z(t)k
for the path in Cartesian coordinates.
(a) Recall that the standard basis vectors for cylindrical coordinates are
e r = cos θ i + sin θ j,
e θ = −sin θ i + cos θ j,
e z = k.
Use the facts that x = r cos θ and y = r sin θ to show that we may write x(t) as
x(t) = r (t) er + z(t) ez .
(b) Use the definitions of er , eθ , and ez just given and the chain rule to find der /dt, deθ /dt, and dez/dt in terms of er , eθ , and ez .
(c) Now use the product rule to give expressions for v and a in terms of the standard basis for cylindrical coordinates.
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