Uniform Circular
Motion
Prepared by: Engr. Odiña
Uniform Circular Motion
Uniform circular motion is the motion of an object
traveling at a constant speed on a circular path.
Uniform Circular Motion
Properties of Uniform Circular motion
1. The speed is constant.
2. The direction of the motion is continually and
uniformly changing.
3. The acceleration is constant in magnitude and
is directed toward the center of the circular
path.
Period
•
Where:
r = radius
T = time (Period)
v = velocity
Period
The wheel of a car has a radius of 0.29m and it
being rotated at 830 revolutions per minute on a
tire-balancing machine. Determine the speed at
which the outer edge of the wheel is moving.
Foundamental Equation
•
Fundamental Equation of Uniformly
Accelerated Angular motion
•
Fundamental Equation of Uniformly
Accelerated Angular motion
•
Where:
= Angular velocity final (rad/s)
= Angular velocity initial (rad/s)
= Displacement (rad)
t = time (s)
RADIAN
1 radian =
57.3 degrees
RELATIONSHIP:
•s
=r
a=r
v=r
STATICS
DYNAMICS
S = Linear distance = Angular distance
V = Linear velocity = Angular velocity
a = Linear Acceleration = Angular Acceleration
Example 1:
A flywheel is rotating at a constant rate of 2400
rpm. At what constant rate in rad/s^2 must it’s
motion be retarded to bring it to rest in 300
revolutions?
a. 16.86
b. 17.86
c. 16.76
d. 18.67
Example 2:
A car has wheels of radius 30cm. It starts from rest
and accelerates uniformly to a speed of 15 m/s in a
time of 8s. Find the number of rotations one wheel
makes in this time.
a. 25 rev
b. 32 rev
c. 45 rev
d. 23 rev
Example 3:
An electric fan revolving at 900 rpm slows down to
300 rpm while making 50 revolutions. Find the
time required to turn this 50 revolutions.
a. 3s
b. 4s
c. 5s
d. 6s
Example 4:
It takes 3 seconds for the shaft of a motor to attain
its full speed of 300rpm. If it started from rest, find
the angular acceleration of the shaft.
a. 5 rad/s^2
b. 10.47 rad/s^2
c. 100 rad/s^2
d. 1.67 rad/s^2
Central Acceleration
"The centripetal acceleration is the rate of
change of tangential velocity."
When an object is moving with uniform
acceleration in circular direction, it is said to be
experiencing the centripetal acceleration.
Central Acceleration
•
Where:
= central acceleration
v = linear speed
r = radius
= angular speed
Example 5:
A 5-kg object moves at a constant speed of 10 m/s
in a 5.0 m radius circle. What is the object's
acceleration?
a. 19 m/s^2
b. 20 m/s^2
c. 21 m/s^2
d. 22 m/s^2
Example 6:
A test car moves at a constant speed around a
circular track. If the car is 48.2 m from the center of
track and has a centripetal acceleration of 8.05
m/s2, what is the car’s tangential speed?
a. 19.69 m/s
b. 69.19 m/s
c. 99.66 m/s
d. 66.99 m/s
Example 7:
A space station is in a circular orbit about the Earth
at an altitude h of 5.0x10^2 km. If the station
makes one revolution every 95min, what are its (a)
orbital speed and (b) centripetal acceleration?
a. 7.6x10^3 m/s, 8.4 m/s^2
b. 6.7x10^3 m/s, 4.8 m/s^2
c. 7.6x10^4m/s, 8.4 m/s^2
d. 6.7x10^3 m/s, 4.8 m/s^2
Centrifugal Force
•
The
inward force that causes the central
acceleration.
= Centripetal force
m = mass
v = linear speed
= angular speed
r = radius
Total Acceleration:
•
Where:
= Total acceleration
= Central acceleration
a = linear acceleration
Banking curves
•
The
proper banking of a curve to eliminate the
necessity for a sidewise frictional force is given by
the relation
Where:
v = linear velocity
r = radius
= angle of banking
Banking curves
Example 8:
Determine the angle of super elevation of hi-way
curve of 183m radius so that there will no side
thrust for a speed of 72kph.
a. 12.56 degrees
b. 12.32 degrees
c. 25.36 degrees
d. 13.66 degrees
Example 9:
A highway curve in Taguig has a radius of 160m.
The curve is banked so that a car travelling at
25m/s will not skid sideways, even if the curve is
coated with a frictionless glaze of ice. At what
angle to the horizontal is the curve banked?
a. 18.3 degrees
b. 21.7 degrees
c. 20.5 degrees
d. 24.6 degrees