Question

# Your task will be to derive the equations describing the velocity and acceleration in a polar...

Your task will be to derive the equations describing the velocity and acceleration in a polar coordinate
system and a rotating polar vector basis for an object in general 2D motion starting from a general
position vector. Then use these expressions to simplify to the case of non-uniform circular motion, and
finally uniform circular motion.
Here's the time-dependent position vector in a Cartesian coordinate system with a Cartesian vector
basis: ⃗r(t)=x (t)
̂
i+y(t)
̂
j where x(t) and y(t) are scalar functions of time (that can be positive or
negative) and describe the components of ⃗r(t) pointing in the fixed unit vector directions ̂
i and ̂
j ,

respectively.
Here's the position vector in polar coordinates but still with fixed ̂
i and ̂
j unit vectors:

⃗r(t)=r(t)cos(θ(t))

̂i+r(t)sin(θ(t))

̂j (eq. 1) where r(t) and θ(t) are scalars representing the distance
from the origin to the object and the angle of the position vector measured counter-clockwise (CCW)
from the ̂
i direction, respectively. For an object with a general 2D trajectory these are both functions of
time.
Now let's swap out our fixed ̂
i and ̂
j vector basis for the more convenient rotating polar vector basis:

Let r̂(t)=cos(θ(t))
̂
i+sin (θ(t))
̂
j (eq. 2) be the radial unit vector. At all times r̂ (t) points from the

origin to the object and has a magnitude of 1. Let θ̂(t)=−sin(θ(t))

̂i+cos(θ(t))

̂j (eq. 3) be the
tangential unit vector. At all times it points 90 degrees CCW from the r̂ (t) direction and has a
magnitude of 1.
When we switch to a polar vector basis, the position vector is rewritten as: ⃗r(t)=r(t)̂r(t) (eq. 4)
Equation 4 will be your starting point. Take its derivative with respect to time once to get the velocity
equation, and then take another derivative to get the acceleration equation. I would recommend
dropping the (t)'s and using a dot above the variable to stand for the time derivative to save space. Two
dots stands for second derivative, for example: r ̇=
dr
dt , r ̈=
d
2
r
dt 2

a) Evaluate and sketch the unit vectors from equations 2 and 3 (with their tails at the origin) for an
object at θ=0,π/4, π/2,π
b) Take derivatives of equations 2 and 3 to prove that ̇r̂=θ ̇ θ̂ and ̇
θ̂ =−θ ̇ r̂
c) Take the derivative of equation 4 to show that ⃗v (t)=r ̇(t)̂r+r(t)ω(t)θ̂ (eq. 5)
d) Take the derivative of equation 5 to show that ⃗a(t)=[r ̈(t)−r(t)ω
2
(t)]r̂+[2r ̇ (t)ω(t)+r(t)α(t)]θ̂

(eq. 6)
(where ω(t) is the object's angular velocity in [rad /s] and α(t) is the object's angular acceleration in
2
] )
e) Show that equations 5 and 6 simplify to the equations in your notes for non-uniform circular motion.
Explicitly state the condition that makes this simplification. (What defines circular motion?)
f) Show that equations 5 and 6 simplify to the equations in your notes (or your textbook) for uniform
circular motion. Explicitly state the conditions that make this simplification. (What defines uniform
circular motion?)

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