Question

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

Please complete the following:

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

[rad /s

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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Answer #1

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