CHAPTER 6: Circular motion (3 Hours)

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CHAPTER 6 Circular motion(3 Hours)
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LEARNING OUTCOME
6.1 Uniform circular motion (1 hour)

• At the end of this chapter, students should be
  able to
• Describe graphically the uniform circular motion
  in terms of the change in direction of velocity.

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6.1 UNIFORM CIRCULAR MOTION
Circular motion motion which occurs when
bodies rotate in circular path.
(Vertical plane)
(Horizontal plane)

• Examples
• a ball is swung in horizontal circle.
• a ca /motorcycle turning a corner.
• cone pendulum.
• merry go round.

• Examples
• a bucket of water is swung in vertical circle.
• roller coaster cars

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6.1 Uniform Circular Motion
Uniform circular motion is the motion of an
object in a circle with a speed.

• . of its velocity remains constant.
• . of its velocity changes continually.

Comparison Of Linear And Circular Motion
Linear motion Circular motion
Displacement, s (m) Angular displacement, ? (rad)
Velocity, (m/s) Angular velocity, (rad/s)
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6.1 Uniform Circular Motion
MOTION CHARACTERICTICS FOR CIRCULAR MOTION
Linear Distance ( )
The arc length between A and B
Angular Displacement ( )
The angle subtended by the arc length. Unit
radian ( rad )
Relation between , r,
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6.1 Uniform Circular Motion
Motion Characterictics For Circular Motion
ANGULAR VELOCITY ( ? )

• Rate of change of angular displacement.

angular displacement (rad)
time taken (s)
- Unit ?
- Other units.
- and its direction is perpendicular to
the plane of motion (right hand rule)
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Motion Characterictics For Circular Motion
6.1 Uniform Circular Motion
THE RELATIONSHIP BETWEEN LINEAR VELOCITY, V AND
ANGULAR VELOCITY, ?
Divide both sides by ,
- The direction of linear velocity at every point
along the circular path is ..to the point.

• The direction of the angular velocity, ? depends
  
• of the object (clockwise or counterclockwise ).

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• Period, T
• is defined as the ..revolution
  (cycle/rotation).
• The unit of the period is ...
• Frequency, f
• is defined as the (cycles/rotations)
  completed in one second.
• The unit of the frequency is .or ..
• Equation
• Let the object makes one complete revolution in
  circular motion, thus
• the distance travelled is (circumference
  of the circle),
• the time interval is one period, T.

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Motion Characterictics For Circular Motion
6.1 Uniform Circular Motion

LINEAR VELOCITY can be written in terms of
period, T and frequency, f
ANGULAR VELOCITY can be written in terms of
period, T and frequency, f
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Motion Characterictics For Circular Motion
6.1 Uniform Circular Motion
EXAMPLE 6.1.1
An object undergoes circular motion with uniform
angular speed 100 rpm. Calculate (a) the
period, T (b) the frequency of revolution, f.
SOLUTION 6.1.1
Given ? 100 rpm Convert to rad s-1
(a)
(b)
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Motion Characterictics For Circular Motion
6.1 Uniform Circular Motion
EXAMPLE 6.1.2
An object travels around the circumference of
a circle of radius 6 m at a rate of 30 rev/min.
Calculate (a) its angular speed in rad/s.
(b) its linear speed around the circle.
SOLUTION 6.1.2
r 6m ? 30 rev/min
(a)
(b)
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EXAMPLE 6.1.3
The diameter of a tire is 64.8 cm. A tack is
embedded in the tread of the right rear tire.
What is the magnitude and direction of the
tack's angular velocity vector if the vehicle is
traveling at 10.0 km/h?  
Solution 6.1.3  
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Learning Outcome
6.2 Centripetal force (2 hours)

• At the end of this chapter, students should be
  able to
• Define and use centripetal acceleration
• Define and solve problems on centripetal force

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6.2 Centripetal Acceleration, ac

• When an object moving in a circle of radius, r
  at a ., v, the direction of the object
  .... Thus it has an acceleration called the
  centripetal acceleration.

• ac is defined as the acceleration of an object
  moving in circular path and it directed
  .of the circle.

• Direction of ac graphically.

FIGURE 6.2.1
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6.2 Centripetal Acceleration, ac
speed of the object
Magnitude of ac
radius
v linear tangential velocity ? angular
velocity (angular frequency) r radius of
circular path
since v r ?,
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Step Technique Explanation
1 Drawing diagram. Draw proper diagram especially the path of the circular motion).
2 Identifying and drawing external forces. Identify and draw all the external forces (including applied and reaction forces) acting on the object. Make sure - the direction of all forces is correct and labeled. - the number of forces is correct. - the tip of all forces is concentrated at the same point (called the origin point).
3 Resolving external forces into components. Resolve all the external forces into x-axis (called x-components) as well as y-axis (called y-components)
4 Drawing centripetal force. Draw the centripetal force, Fc which is always pointing to the centre of the circular path.
5 Identifying the type of motion. Identify whether the circular motion is horizontally or vertically.
6 Calculation. Use the correct condition to relate the centripetal force with the all the external forces of the same axis. Assign the external forces as positive if the direction is the same as the direction of the centripetal force. negative if the direction is opposite to the direction of the centripetal force.
Uniform Circular Motion Problem Solving
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6.2 Centripetal Acceleration, ac
Example 6.2.1
Calculate the centripetal acceleration of a car
traveling on a circular racetrack of 1000 m
radius at a speed of 180 km h-1.
Solution
Given r 1000 m v 180 km h-1
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Centripetal Force, Fc
Fc is defined as the ..required to keep an
object of mass, m moving at a speed v on a
circular path of radius, r.
Examples
As the moon orbits the Earth, the force of
gravity acting upon the moon provides the
centripetal force required for circular motion.
FIGURE 6.2.2
As a car makes a turn, the force of friction
acting upon the turned wheels of the car provides
the centripetal force required for circular
motion.
FIGURE 6.2.3
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Centripetal Force, Fc
Magnitude of Fc
since v r ? , thus
Direction of Fc - of the circle and
..of the centripetal acceleration.
Fc is perpendicular to the direction v, so it
does no work on the object.
FIGURE 6.2.4
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Circular motion in horizontal plane (uniform
circular motion)
Case 1 object moves in a horizontal circle with
steady speed.
T
mg
FIGURE 6.2.5

• Two forces acting on the object
• The force of .( weight i.e. mg )
• The .in the string - is the only component
  in the radial direction that provided the
  centripetal force.

Applying Newton 2nd Law
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Circular motion in horizontal plane (uniform
circular motion)
Example 6.2.2
A 0.25 kg rock attached to a string is whirled in
a horizontal circle at a constant speed of 10.0
ms-1. The length of the string is 1.0m.
Neglecting the effects of gravity, find the
tension in the string.
Solutio 6.2.2
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Case 2 Motion of car round a curve Flat curve
road
If the coefficient of static friction between the
tires the road is µ then,
f s µR
Vertical comp.
Horizontal comp.
( 2 )
( 1 )
(1) into (2)
? The maximum velocity without slipping on the
road is
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Motion of car round a curve
Example 6.2.3
A car travels around a flat curve of radius r
50m. The coefficient of the static friction
between the tires the road is µs 0.75.
Calculate the maximum speed at which the car can
travel without skidding.
Solution 6.2.3
Given
µs 0.75 , r 50m
fs supplies the centripetal force
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Motion of car round a curve
Example 6.2.4
A 1200 kg car with a velocity of 8.0 m/s travels
around a flat curve of radius r 9.0m. a)
Calculate the horizontal force must the
pavement exert on the tires to hold the car
in the circular path ? b) What coefficient of
friction must exist for the car not to slip
?
Solution 6.2.4
a)
b)
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Case 3 Conical Pendulum
A conical pendulum moving in uniform circular
motion with speed v
T sin ? supplies centripetal force.
r L sin ?
Component x
r
FIGURE 6.2.7
Component y
(1)
(2)
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Case 3 Conical Pendulum
r L sin ?
(1)
(2)
r
FIGURE 6.2.8
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Conical Pendulum
Example 6.2.6
A 0.15 kg ball attached to a string which is 1.2
m in length moves in a horizontal circle. The
string makes an angle of 30 with the vertical.
Find the tension in the string the speed of the
ball.
Solution 6.2.6
Component - x
Component - y
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CIRCULAR MOTION IN VERTICAL PLANE (NON-UNIFORM
CIRCULAR MOTION)
Case 1 A ball is attached to a string moves
in a vertical circle.
FIGURE 6.2.9
FIGURE 6.2.10
At the top of the circle ( point A ) both T
mg are directed downwards
At the bottom of the circle ( point B ) T mg
point in opposite direction
(T is minimum)
(T is maximum)
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Circular motion in vertical plane (non-uniform
circular motion)
Circular motion is possible as long as the cord
remain taut, thus there is a critical (minimum)
speed to be maintained.
If the rope is sagging, T 0 , thus
T0
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Circular motion in vertical plane (non-uniform
circular motion)
Example 6.2.7
constant speed
A 1.2 kg rock is tied to the end of a 90 cm
length of string. The rock is then is whirled in
a vertical circle at a constant speed of 8 m/s.
What are the tensions in the string at the top
and bottom of the circle ?
Solution 6.2.7
m 1.2 kg , r 90 cm , v 8 m/s
Top
Bottom
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Circular motion in vertical plane (non-uniform
circular motion)
Example 6.2.8
not constant speed
A 2 kg ball is tied to the end of a 80 cm
length of string. The ball is then is whirled in
a vertical circle and has a velocity of 5 m/s at
the top of the circle. a) What is the tension
in the string at that instant ? b)
What is the minimum speed at the top necessary to
maintain circular motion ?
Solution 6.2.8
m 2 kg , r 80 cm , v 5 m/s at the top.
a)
b)
Top
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Circular motion in vertical plane (non-uniform
circular motion)
Example 6.2.9
A rope is attached to a bucket of water and the
bucket is then rotated in a vertical circle of
0.70 m radius. Calculate the minimum speed of the
bucket of water such that the water will not
spill out.
Solution 6.2.9
The water will not spill out if the T0, thus
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Circular motion in vertical plane (non-uniform
circular motion)
Case 2 Roller coaster on a circular track /
Ferris wheel
Top of the circle
Bottom of the circle
R
R
mg
mg
A minimum velocity (when R 0) is required in
order to keep a roller coaster car on a circular
track.
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Circular motion in vertical plane (non-uniform
circular motion)
Example 6.2.10
What minimum speed must a roller coaster be
traveling when upside down at the top of a
circle (refer to the figure) if the passengers
are not to fall out ? Assume R 8.0 m.
Solution 6.2.10
r 8.0 m
Top
For vmin , R 0
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Example 6.2.11
A rider on a Ferris wheel moves in
a vertical circle of radius, r 8 m at
constant speed, v as shown in Figure 6.13. If the
time taken to makes one rotation is 10 s and
the mass of the rider is 60 kg, Calculate the
normal force exerted on the rider a. at the top
of the circle, b. at the bottom of the
circle. (Given g 9.81 m s-2)
Figure 6.13
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Solution 6.2.11 a. The constant speed of the
rider is The free body diagram of the
rider at the top of the circle
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Solution 6.2.11 b. The free body diagram of the
rider at the bottom of the circle
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Motion Characterictics For Circular Motion
6.1 Uniform Circular Motion
EXERCISES 6.1

1. A particle is moving on a circular path of
   radius 0.5 m at a constant speed of 10 m/s.
   Calculate the time taken to complete 20
   revolutions.

(t 6.28 s)
2. Two wheels of a machine are connected by a
transmission belt. The radius of the
first wheel r1 0.5 m, the radius of the second
wheel r2 0.125 m. The frequency of the
bigger wheel equals 3.5 Hz. What is the frequency
of the smaller wheel ?
(f2 14 Hz)
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Motion Characterictics For Circular Motion
6.1 Uniform Circular Motion
3. The astronaut orbiting the Earth is preparing
to dock with Westar VI satellite. The satellite
is in a circular orbit 600 km above the Earths
surface, where the free fall acceleration is 8.21
m s?2. Take the radius of the Earth as 6400 km.
Determine a. the speed of the satellite, b.
the time interval required to complete one orbit
around the Earth. ANS. 7581 m s?1 5802 s
4. A pendulum bob of mass 1 kg is attached to a
string 1 m long and made to revolve in a
horizontal circle of the radius 60 cm. Calculate
the period of the motion and the tension
of the string.
Tension , T 12.25 N