Download Circular Motion

Which horse has a
larger linear speed
on a merry go round,
one on the outside or
one on the inside?
Outside.
Which horse has a
greater rotational
speed?
Neither, all the horses
complete the circle in
the same amount of
time.
Circular Motion
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How much faster will a horse at TWICE
the distance from the center of the
circle be moving
Rotating
Turning about an
internal axis
Revolving
Turning about an
external axis
TWICE the distance means TWICE the
speed
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Linear speed
How far you go in a certain amount of time
Miles per hour, meters per second
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The number of revolutions per second is called the
frequency, f.
Frequency is measured in Hertz, Hz.
The time it takes to go all the way around once is
called the period, T.
Frequency is related to period by
f=1/T
and
T=1/f
Rotational speed
How many times you go around in a certain
amount of time
Revolutions per minute, rotations per hour
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Example
Uniform Circular Motion, UCM: moving in a circle
with a constant speed.
Question: Is there a constant velocity when an
object moves in a circle with a constant speed?
No, the direction changes, therefore the velocity
changes.
If the velocity changed, the object is actually
ACCELERATING even while moving at the
same speed.
A boy twirls a toy airplane around and
around at the end of a string. If it takes
2 seconds for the airplane to complete
one loop, what is the frequency?
f=1/T
f=1/2
f = 0.5 Hz
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How do you find the velocity of an object
moving in a circle if it is not directly
provided?
We know that Velocity = distance / time
In circular motion, the distance traveled is all
around the circle4 the circumference.
The circumference = 2πr
The time it takes the object to go all the way
around the circumference once is called
the period, T.
So4
v = 2πr / T
Suppose an object was moving in a straight line with some
velocity, v.
According to Newton’s 1st Law of Motion, “An object in motion
continues that motion unless a net external force acts on
it”.
If you want the object to move in a circle, some force must
push or pull it towards the center of the circle.
Example
A race car takes 1.5 minutes to go around
one lap of a circular track. If the track has
a radius of 400 m, how fast was the car
traveling?
v = 2πr / T
v = 2π(400) / (1.5 x 60)
v = 27.9 m/s
A force that pushes or
pulls an object towards
the center of a circle is
called a centripetal force
Centripetal means “center
seeking”
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What happens is the string breaks?
Which way will the ball move?
Centripetal force
Since Fnet = ma, the net centripetal force is
given by
Fnet
The ball will continue to move in a
straight line path that is “tangent”
to the circle.
v2
=m
r
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• Centrifugal is Center – fleeing (away from
the circle)
Lots of forces can help in pushing or pulling
an object towards the center of a circle.
Sometimes it takes more than one force to
get an object to move in uniform circular
motion.
Centripetal force is NOT a new kind of force.
If any force is making an object move in a
circle, it becomes a centripetal force.
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When can these forces be
centripetal forces?
Gravity?
Moon revolving around the
Earth
Tension?
Twirling a pail at the end of
a string
Friction?
Cars rounding a curve.
Air Resistance?
Birds flying in a circle.
Normal?
Riders in a carnival ride
According to Newton’s 2nd Law, Fnet = ma,
If there is a centripetal force, there must
be a centripetal acceleration.
ac = v2 / r
Where r is the radius of the circle and v is
the velocity of the object.
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Example
Friction
A boy twirls a ½ kg rock in a horizontal circle on the end
of a 1.6 meter long string. If the velocity of the rock was
4 m/s, what is the Tension in the string?
m = ½ kg
r = 1.6 m
v = 4 m/s
v2
The only centripetal force is Tension. Fnet = m
r
T = m v2 / r
2
T = ½ 4 / 1.6
T=5N
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Example
A 1500 kg race car goes
around a circular track
at 45 m/s. If the radius
of the track is 100 m,
how much friction is
require to keep the car
on the track?
m = 1500 kg
v = 45 m/s
r = 100 m
The centripetal force is
friction.
friction = m
v2
r
How much more friction is
required if the velocity doubles?
Triples?
What happens to the friction if
the radius is doubled? Halved?
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Friction
How fast was the ½ kg rock moving if the Tension
was 10 N and the string was 1.6 m long?
m = ½ kg
r = 1.6 m
T = 10 N
T = mv2 / r
Tr/m = v2
10 x 1.6 / .5 = v2
v = 5.7 m/s
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A 1500 kg race car goes
around a circular track at 45
m/s. If the radius of the
track is 100 m, how much
friction is require to keep the
car on the track? What is µ,
the coefficient of friction?
m = 1500 kg
v = 45 m/s
r = 100 m
The centripetal force is friction.
v2
Fnet = m
r
f = mv2/r
f = 1500 x 452 / 100
f = 30375 N
f = µN
µ= f / N
N = mg = 15000 N
µ = 30375 N / 15000 N
µ = 2.02
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Example
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On a rotating space station with a radius = 20 m, a point on
the rim is moving at 12 m/s. What is the centripetal
acceleration? How heavy would a 70 kg astronaut
feel?
ac= v2/r
ac= 122/20
ac = 7.2 m/s2
Since the centripetal acceleration, ac
provides the “simulated gravity, g”
He “feels” a weight of not mg, but mac.
Apparent weight = 70 x 7.2 = 504 N
This is somewhat less than his real weight of 700 N, but
still feels more “normal” than the sensation of
“weightlessness”
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Loop the Loop
What is the minimum speed that a rider must be moving at
in order to complete a loop the loop of radius 12 m?
The weight is the only centripetal force when the rider is
moving at the minimum required speed.
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Fnet = m
v
r
mg = mv2/r
g = v2/r
v2 = rg
v2 = 12 x 10
v = 10.95 m/s
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“Artificial Gravity”
Occupants of a space station feel weightless because they lack a
support (Normal) force. By spinning the station as just the right
speed, they will experience a “simulated gravity” when the Normal
force of the floor pushing on them becomes a centripetal force. The
closer their centripetal acceleration, v 2/r is to g, the acceleration due
to Earth’s gravity, the more they feel the sensation of normal weight.
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