In this problem, we will find expressions for velocity and acceleration in cylindrical c...

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) e_r + z(t) e_z .

(b) Use the definitions of e_r , e_θ , and e_z just given and the chain rule to find de_r /dt, de_θ /dt, and de_z/dt in terms of e_r , e_θ , and e_z .

(c) Now use the product rule to give expressions for v and a in terms of the standard basis for cylindrical coordinates.