Circular motion We will be looking at a special case of kinematics and dynamics of objects in uniform circular motion constant speed Cars on a circular track or on a curved road Roller coasters in loop ID: 236132
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Slide1
CIRCULAR MOTIONSlide2
Circular motionWe will be looking at a special case of kinematics and dynamics of objects in uniform circular motion (constant speed)Cars on a circular track (or on a curved road)Roller coasters in loopOther amusement park rides like
ferris
wheels, rotor, scrambler
Rotating objects (like a ball rolling) (these are moving in a circular path even though radius is very small)
Orbits of planets
Running back cutting up fieldSlide3
First let’s consider a mass on a string being twirled overhead at a constant speed.
Let’s determine the speed of the object.
Remember that speed is defined as:Slide4
We define the
period of motion (T)
as the time it takes to complete one rotation.
How far does it travel in one rotation?
We can find the circumference of the circular path by:
Therefore the speed of an object in uniform circular motion is:
So the speed depends on the radius of the circle (think about runners on a track – outside lane must run faster)Slide5
Ok so we’ve figured out its speed, but is it accelerating?
Remember that it is traveling at a constant speed. However, acceleration is defined as:
So how does the
velocity
change with respect to time?
So even though speed is constant, velocity changesSlide6
Which direction is velocity?http://www.youtube.com/watch?v=zww3IIMRo4USlide7
Notice that the direction of the velocity at any time
is tangent to
its path.Slide8
What direction is acceleration
v
v
Acceleration is the change in velocity.
Acceleration is towards center of circle!
v
f
– v
i
– Let’s look at graphicallySlide9
An animation of velocity and accelerationhttp://www.physicsclassroom.com/Physics-Interactives/Circular-and-Satellite-Motion/Uniform-Circular-Motion/Uniform-Circular-Motion-InteractiveSlide10
So even though it may be traveling at a
constant speed
any body traveling in a circular path is
accelerating
because the
direction of its velocity is always changing.
The acceleration of an object in uniform circular motion is:
ac
=
v
2
r Slide11
It is important to note that the direction of the change in velocity is always
towards the center of the circle
.
Therefore the acceleration of an object in circular motion is always towards the center of the circle
. – always!!!This is the definition of centripetal, which means center-seeking.Slide12
tangential accelerationAn object CAN have both tangential and centripetal acceleration. If you drive a car around a curve at 45 km/hr, there is centripetal acceleration. If you speed up (accelerate) to 50 km/
hr
, there is centripetal AND tangential acceleration
We will not consider tangential accelerationSlide13
Whenever an object is accelerated there must be a
net force
acting on it
.
This force is known as centripetal force, Fc. This is not a new force
, it is simply the net force that accelerates an object towards the center of its circular path.Slide14
Examples
:
1.
A mass is twirled in a circle at the end of a string, the centripetal force is provided by…tension
. When a car rounds a corner on a highway, the centripetal force is provided by…friction. When the Moon circles the Earth, the centripetal force is provided by…
gravityOn a FBD, label the centripetal force as specifically as you can (not Fc)Slide15
Newton’s Second Law we can help us to determine a formula for centripetal force:Slide16
Example:
A 0.50 kg mass sits on a frictionless table and is attached to hanging weight. The 0.50 kg mass is whirled in a circle of radius 0.20 m at 2.3 m/s. Calculate the centripetal force acting on the mass.
Calculate the mass of the hanging weight.Slide17
Example:
A car traveling at 14 m/s goes around an unbanked curve in the road that has a radius of 96 m. What is its centripetal acceleration?Slide18
What is the minimum coefficient of friction between the road and the car’s tires in the last question?Slide19
19
Example
A
model airplane has a mass of 0.90 kg and moves at a constant speed on a circle that is parallel to the ground.
Find
the tension T in the guideline(length=17m) for
speed of 19 m/s.
FC=T=mv2/rSlide20
How could you determine the radius of a circle if all you had was an object on a string, a scale, a stopwatch and a force gauge (like a spring scale)Slide21
Example:
A plane makes a complete circle with a radius of 3622 m in 2.10 min. What is the speed of the plane?Slide22
Example. The wall exerts a 600 N
force on an
80-kg
person moving at
4 m/s on a circular platform. What is the radius of the circular path?
r
= 2.13 mDraw and label sketch
r = ?
m =
80 kg;
v
= 4 m/s
2
F
c
=
600 NSlide23
The Conical Pendulum
A
conical pendulum
consists of a mass
m
revolving in a horizontal circle of radius R at the end of a cord of length L.
q
h
T
L
R
mg
T
q
T sin
q
T cos
q
Note: The inward component of tension
T sin
q
gives the needed central force.
http://
www.youtube.com/watch?v=5C4RJlFABicSlide24
Angle q and velocity
v
:
q
h
T
L
R
mg
T
q
T sin
q
T cos
q
T
cos
q
= mg
mv
2
R
T
sin
q
=
Solve two equations to find angle
q
tan
q
=
v
2
gRSlide25
Example : A 2-kg
mass swings in a horizontal circle at the end of a cord of length
10 m
. What is the constant speed of the mass if the rope makes an angle of
300 with the vertical?
R = L sin 300 = (10 m)(0.5)
R = 5 m1. Draw & label sketch.
2. Recall formula for pendulum.
Find:
v = ?
3. To use this formula, we need to find R = ?
q
h
T
L
R
q =
30
0Slide26
Example 6(Cont.): Find v
for
q
= 30
0R = 5 m
v = 5.32 m/s
g = 10 m/s2Solve for v = ?
4. Use given info to find the velocity at
30
0
.
q
h
T
L
R
q =
30
0
R =
5 mSlide27
One last note on a little thing called centri
fugal
force. While centripetal means center-
seeking
centrifugal means center- fleeing
.Centrifugal force is actually an apparent force - it does not exist. It is simply the apparent force that causes a rotating object to move in a straight line.However, Newton’s First Law tells us that we do not need a force to keep an object moving in a straight line, you only need a force to deflect
an object from moving in a straight line.In reality what we seem to feel as centrifugal force is really…Slide28
Example
:
When riding in the backseat of a car that is turning a corner, you slide across the seat, seeming to accelerate outwards, away from the center of the turning circle.
In reality your forward inertia you had before the car started to turn makes you want to continue in a straight line
(which makes you feel like you are sliding out)
When you slide into the side door, it exerts a centripetal force (normal force in this case) and accelerates you towards the center of the turn.