UNIFORM CIRCULAR MOTION
MATHEMATICS OF CIRCLES AND
RADIANS
I. DEPICTING CIRCULAR MOTION
A. MATHEMATICS OF CIRCLES AND RADIANS
B. VECTOR DIAGRAMS OF CIRCULAR MOTION
II. CENTRIPETAL FORCE AND ACCELERATION
A. ACCELERATION TOWARDS THE CENTER OF A CIRCLE
B. EQUATING CENTRIPETAL FORCE TO OTHER FORCES
𝐶 = 2𝜋𝑟 = 𝜋𝑑
A = 𝜋𝑟 2
Angles are measured in degrees (360° in
a circle) and radians (6.28 rad in a circle).
¼ circle = 90° = 1.57 rad
½ a circle = 180° = 3.14 rad
¾ a circle = 270° = 4.71 rad
1 full circle = 360° = 6.28 rad
So….. 1 rad = 57.3° and 1° = 0.0175 rad
UNIFORM CIRCULAR MOTION
Why are there 6.28 radians in a circle?
It is not random….
I. DEPICTING CIRCULAR MOTION
A. MATHEMATICS OF CIRCLES AND RADIANS
B. VECTOR DIAGRAMS OF CIRCULAR MOTION
II. CENTRIPETAL FORCE AND ACCELERATION
A. ACCELERATION TOWARDS THE CENTER OF A CIRCLE
B. EQUATING CENTRIPETAL FORCE TO OTHER FORCES
If s = r, then  = 1 radian (defined)
So, the angle  in radians = arc length
divided by radius
or
𝜽=
𝒔
𝒓
or
𝒔 = 𝜽𝒓
If s = C = 2r, then 𝜽 =
therefore:
𝟐𝝅𝒓
𝒓
360 = 2 rad,
180 =  rad,
90 = /2 rad
...etc……
= 𝟐𝝅 ;
r

s
DEFINE: One radian is the
angle swept out by an arc of
length equal to the radius of
the circle.
UNIFORM CIRCULAR MOTION
If
I. DEPICTING CIRCULAR MOTION
A. MATHEMATICS OF CIRCLES AND RADIANS
B. VECTOR DIAGRAMS OF CIRCULAR MOTION
II. CENTRIPETAL FORCE AND ACCELERATION
A. ACCELERATION TOWARDS THE CENTER OF A CIRCLE
B. EQUATING CENTRIPETAL FORCE TO OTHER FORCES
Source: Calculus, Swokowski
Understand how the sine and cosine functions can be graphed along the
x axis, especially why the units of x-axis are radians.
I. DEPICTING CIRCULAR MOTION
A. MATHEMATICS OF CIRCLES AND RADIANS
B. VECTOR DIAGRAMS OF CIRCULAR MOTION
II. CENTRIPETAL FORCE AND ACCELERATION
A. ACCELERATION TOWARDS THE CENTER OF A CIRCLE
B. EQUATING CENTRIPETAL FORCE TO OTHER FORCES
UNIFORM CIRCULAR MOTION
CONSIDER: An object moving in a
circle, radius R, at constant speed v.
T = time to complete one revolution =
‘period’
Object covers distance 2πR in time T,
2𝜋𝑅
so 𝑣 =
𝑇
In time T, object sweeps out an angle
2π. Therefore,
Source: Physics for the IB Diploma, 5th Ed, Tsokos
‘Angular speed’ =
𝑎𝑛𝑔𝑙𝑒 𝑠𝑤𝑒𝑝𝑡
𝑡𝑖𝑚𝑒 𝑡𝑎𝑘𝑒𝑛
or
𝝎=
‘Angular frequency’ =
(angle swept)(frequency) or
𝟐𝝅
𝑻
units = s-1
ω = 2πf
units = s-1
UNIFORM CIRCULAR MOTION
VECTOR DIAGRAMS OF CIRCULAR
MOTION
I. DEPICTING CIRCULAR MOTION
A. MATHEMATICS OF CIRCLES AND RADIANS
B. VECTOR DIAGRAMS OF CIRCULAR MOTION
II. CENTRIPETAL FORCE AND ACCELERATION
A. ACCELERATION TOWARDS THE CENTER OF A CIRCLE
B. EQUATING CENTRIPETAL FORCE TO OTHER FORCES
v and v are not the same; even though
v is constant, v is constantly changing.
v is always tangent to the circle…. And
Δv always points inwards.
Source: Physics for the IB Diploma, 5th Ed, Tsokos
Therefore, ALWAYS an a acting inwards…. So always a F acting
inwards.
UNIFORM CIRCULAR MOTION
I. DEPICTING CIRCULAR MOTION
A. MATHEMATICS OF CIRCLES AND RADIANS
B. VECTOR DIAGRAMS OF CIRCULAR MOTION
II. CENTRIPETAL FORCE AND ACCELERATION
A. ACCELERATION TOWARDS THE CENTER OF A CIRCLE
B. EQUATING CENTRIPETAL FORCE TO OTHER FORCES
EXAMPLE 1
A mass on a string moves in a circle of radius 3.0 m with a constant speed of v
= 6.0 ms-1. However, v changes constantly (why?) and is at all times tangent to
the circular path as shown. The mass goes from A to B to C. Find:
a) Δv (vB - vA) between A and B
Source: Physics for the IB Diploma, 5th Ed, Tsokos
UNIFORM CIRCULAR MOTION
I. DEPICTING CIRCULAR MOTION
A. MATHEMATICS OF CIRCLES AND RADIANS
B. VECTOR DIAGRAMS OF CIRCULAR MOTION
II. CENTRIPETAL FORCE AND ACCELERATION
A. ACCELERATION TOWARDS THE CENTER OF A CIRCLE
B. EQUATING CENTRIPETAL FORCE TO OTHER FORCES
EXAMPLE 1 (con’t)
A mass on a string moves in a circle of radius 3.0 m with a constant speed of v
= 6.0 ms-1. The mass goes from A to B to C. Find:
b) Δv (vC – vB) between B and C
Source: Physics for the IB Diploma, 5th Ed, Tsokos
UNIFORM CIRCULAR MOTION
I. DEPICTING CIRCULAR MOTION
A. MATHEMATICS OF CIRCLES AND RADIANS
B. VECTOR DIAGRAMS OF CIRCULAR MOTION
II. CENTRIPETAL FORCE AND ACCELERATION
A. ACCELERATION TOWARDS THE CENTER OF A CIRCLE
B. EQUATING CENTRIPETAL FORCE TO OTHER FORCES
HWSP #1
Estimate the linear speed of an object at the equator due to the rotation of the
Earth.
[470 ms-1]
UNIFORM CIRCULAR MOTION
I. DEPICTING CIRCULAR MOTION
A. MATHEMATICS OF CIRCLES AND RADIANS
B. VECTOR DIAGRAMS OF CIRCULAR MOTION
II. CENTRIPETAL FORCE AND ACCELERATION
A. ACCELERATION TOWARDS THE CENTER OF A CIRCLE
B. EQUATING CENTRIPETAL FORCE TO OTHER FORCES
EXAMPLE 2
Repeat HWSP #1 at two other locations:
a) 60° latitude
b) 90° latitude.
Source: montgomerycollege.edu