We commonly use
Cartesian or rectangular coordinate system where (x, y) identifies a point in two dimensions.
We could equivalently use polar coordinates to identify a point (r, θ)

Given r and θ, Given x and y,
X = r·cosθ
Y = r·sinθ r = (x 2 + y 2 )
½
θ = tan -1 (y/x)
This converts from a polar coordinate to a rectangular coordinate
This converts from a rectangular coordinate to a polar coordinate

Imagine a point moving along the circle at constant speed.
Which variables change and which stay the same?

The angle, θ, changes but the radius does not change.
The coordinates x- and yboth change.
The speed does not change but the velocity does change (why?)

Angles may be measured in degrees or in radians.
360 ° = 2π
The length of an arc subtending a circle of radius r is given by
S = r ·θ

A spectator is standing at the center of a circular running track and observe a runner start practice
256 m due east of her position. The runner runs on the track in a counter clockwise direction to the finish line, located due north of the observer. What was the distance of the run?
A sailor sights a distant ship and finds that it subtends an angle of 1.15
°. He knows from the shipping charts that the tanker is 150 m in length. How far away is the tanker?

Angular speed, ω, is defined as the rate of change of θ.
ω = Δθ/ Δt [radians/sec] and θ = ωt
Then s = rθ becomes s = r (ωt)
And v = r ω

Angular velocity is denoted with ω, omega, and the direction is at a right angle to the plane of the circular motion (use right hand rule)
Tangential velocity is denoted with v, velocity.

Note that ω is measured in radians /sec.
The formulas involving ω are true only if θ
(angular displacement) is measured in radians since only then is s = rθ a true statement.

The Period, T, is the time necessary for one complete cycle. Period, T, is measured in
[seconds].
The frequency, f , is the number of cycles that happen in one second. Frequency is measured in [cycles/second] or [Hertz]
T = 1/ f
Angular speed, ω = 2π/T = 2π f

A merry go round at its constant operational speed makes one complete rotation in 45 seconds. Two children are on horses, on at 3.0 m from the center and one 6.0 m from the center. Find the angular speed and the tangential speed for each child.
A CD rotates in the player at a constant speed of
200 rpm. What are the CD’s a) frequency and b)
Period of revolution?

An object in Uniform Circular Motion
(constant speed) experiences a
Centripetal Force ( and therefore a centripetal acceleration) that is pointed towards the center.
a c
= v 2 /r = (r ω) 2 /r = r ω 2
F c
= m(v 2 /r) = mr ω 2

A lab centrifuge operates at a rotational speed of 12,000 rpm. A) What is the magnitude of the centripetal acceleration of a red blood cell at a radial distance of
8.0 cm from the centrifuge’s axis of rotation? Compare this with g.
A ball attached to a string is swung with uniform motion in a horizontal circle over a person’s head. If the string breaks, what will be the trajectory of the ball?

A car approaches a level circular curve with radius of 45.0 m. If the concrete pavement is dry, what is the maximum (constant) speed at which the car can negotiate the curve?
Two masses, m
1
= 2.5 kg and m
2
= 3.5 kg are connected by light strings and are in uniform circular motion on a horizontal frictionless surface, with r
1
= 1.0m and r
2
= 1.3m. The forces acting an the masses are T
1 and T
2 respectively. Find the centripetal acceleration and the tangential speed of mass 2 and mass 1.

When circular motion isn’t uniform, angular velocity changes.
α , alpha, is used to denote angular acceleration.
We consider only the case of constant angular acceleration so
α av
= α = Δω/t
Then Δω = αt and ω f
= ω i
+ αt

If the rate of angular velocity changes, then the object undergoes angular acceleration and tangential acceleration: a t
= Δv/t = Δ(rω)/t = rα a t
= rα

X = v av t
θ =
ω av t
V av
= (v f
+ v i
)/2
V f
= v i
+ at
X f
= x i
+ v i t + ½ at 2
V f
2 = v i
2 + 2a Δx
ω av
= (ω f
+ω i
)/2
ω f
= ω i
+ αt
θ f
=θ i
+ ω i t + ½ αt 2
ω f
2
=
ω i
2 + 2αΔθ

A CD accelerates uniformly from rest to its operational speed of 500 rpm in 3.0 sec. What is the angular acceleration of the CD during this time? If the CD comes to a stop in 4.0 sec, what is the angular acceleration during that part of the motion?
A microwave oven has a 30 cm rotating plate.
The plate accelerates from rest to a uniform rate of 0.87 rad/s2 for 0.50 sec before reaching its constant operational speed. How many revolutions does the plate make before reaching its operational speed? What are the operational angular speed and tangential speed at its rim?

Newton wondered whether the force pulling apples to the ground was the same force pulling the moon towards the Earth.
He recognized that, although the moon orbits the Earth in a circular path, that the moon really is ‘falling’ towards the Earth

Is always attractive
Is proportional to the product of the masses
Is inversely proportional to the square of the distance between the masses
G = 6.67 X 10 -11 Nm 2 /kg 2

Compute the gravitational force between
Earth and the Moon. M m
M
E
= 6.0 X 10 24 kg, r
EM
= 7.4 x 10 22
= 3.8 X 10 8 m.
kg,
Find the acceleration due to gravity acting on an object with mass M surface. r
E
= 6.4 X 10 6 m o near the Earth’s

Some communication and weather satellites are launched into circular orbits above the Earth’s equator so they are synchronous with Earth’s rotation. At what altitude are these satellites?
R
E
= 6.4 X 10 6 km;
M
E
= 6.0 X 10 24 kg,
T = 365 days