October 2014
Uniform Circular
Motion
Uniform circular motion
What does the word “uniform” mean here?
constant radius and constant speed
Velocity vector is tangent to the path at each instant, so direction of
velocity vector changes all the time as the object moves in circle.
Therefore, there must be an inward-directed acceleration
that causes the velocity to change its direction.
Circular Motion
𝑎 is towards the center.
𝑣 is tangential to the motion
Speed is constant, 𝑣 changes
𝐴 force directed towards the
center is what causes the
acceleration (e.g. gravity,
a string)
If the force is removed, the
ball will continue in a straight
line at the speed it had.
Check your understanding …
When a wheel rotates about a fixed axis, do all the
points on the wheel have the same tangential
speed?
Yes!
Do they all have the same velocity?
No!
Check your understanding …
Circular Motion Equations
vt= 2πr/T
ac = vt2/r
Where
vt = tangental velocity
R = radius
T = period (time required to make one
complete circle)
ac = centripetal acceleration
Circular Motion– We do
The radius of a spacecraft orbiting earth is 6.67 x
106 m. If it orbits earth in 5292 seconds, what is
the velocity of the spacecraft?
vt= 2πr/T = 2*π*6.67X106m / 5292 sec = 7920
m/s
Circular Motion– We do
Jimmie Johnson is driving his #48 Lowe’s NASCAR
around a bend that has a radius of 70 meters. It
takes him 30 seconds to travel the track. What was
the centripetal acceleration of Jimmie John’s #48
Lowe’s NASCAR?
Strategy: find vt, then find ac.
vt= 2πr/T = 2π*70m/60sec = 7.32m/s
ac = vt2/r = (7.32 m/s)2 / 70 m = 0.8 m/s2
NOTICE! T is time to go around a full
circle. Jimmie went ‘around a bend’
which is half a circle. Thus, we need to
double his time to get T.
Centripetal Acceleration – You do
a. A girl sits on a tire that is attached to an overhanging tree limb by a rope. The
girl’s father pushes her so that her centripetal acceleration is 3.0 m/s2. If the length
of the rope is 2.1 m, what is the girl’s tangential speed?
2.5 m/s
b. A boy swings a yo-yo horizontally above his head so that the yo-yo has a
centripetal acceleration of 1.5m/s2. If the yo-yo’s tangential speed is 1.1m/s, what
is the length of the yo-yo?
0.81 m
c. Correct the following statement: The racing car rounds the turn at a constant
velocity of 145 km/h.
Constant speed not constant velocity
Dynamics of uniform circular motion
Object undergoing uniform circular motion is accelerating with
centripetal acceleration ac , so it has a force acting upon it:
mv 2
Fc = mac =
r
which must be directed toward the center of the circle.
It is called centripetal force.
it is not separate force – it is simply one of our familiar
forces acting in the role of causing circular motion
Many forces can force an object to move in circular path,
therefore becoming centripetal force:
Moon around the Earth ………
gravity
object sitting on a rotating table
(strawberries sitting on a seat of a turning car)
(a car moving in a circular path) …………
a ball whirling in a circle
at the end of a string
………
friction
tension in the string
a person pressed against the inner wall of a rapidly rotating circular
room in an amusement park …………. normal force from the wall
Ferris–wheel rider passes through
the lowest point of the ride …….
normal force from the seat
and the force of gravity
useful relations: - speed(linear) and angular speed
 period T:
time required for one complete revolution (s)
speed v = distance/time
constant speed
2πR
v=
T
angular speed ω = angle swept/time
ω=
2π
T
Centrifugal (center-fleeing) Force - MISCONCEPTION
Circular Motion The radius of a spacecraft orbiting earth is 6.67 x 106 m. If it orbits earth in
5292 seconds, what is the velocity of the spacecraft?
7919 m/s
Jimmie Johnson is driving his #48 Lowe’s NASCAR around a bend that
has a radius of 70 meters. It takes him 30 seconds to travel the track.
What was the centripetal acceleration of Jimmie John’s #48 Lowe’s
NASCAR?
3.07 m/s2
When a wheel rotates about a fixed axis, do all the points on the wheel
have the same tangential speed?
Yes
Do they all have the same velocity?
No
A 1000 kg car is going around a curve with radius 30 meters. If the
coefficient of friction between the car's tires and the road is 0.5, what is
the maximum speed at which the car can make the turn?
m = 1000 kg
g = 9.8 m/s2
r = 30 m
𝜇 = 0.5
maximum speed in the turn, v = ?
𝐹𝑐 = 𝐹𝑓𝑟
v = 12 m/s
𝑣2
𝑚
= 𝜇𝐹𝑛
𝑟