Download Uniform Circular Motion

Uniform Circular Motion
Pg. 114 - 130
Uniform Circular Motion
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Have you ever ridden on the ride shown below? As it
spins you feel as though you are being pressed tightly
against the wall. And then the floor drops away and the
ride begins to tilt. But you remain “glued” to the wall.
What is unique about moving in a circle that allows you
to apparently defy gravity? What causes people on the
ride to “stick” to the wall?
Uniform Circular Motion
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Amusement park rides are only one of a very large
number of examples of circular motion. When an object
is moving in a circle and its speed is constant, it is said to
be moving with uniform circular motion
Uniform Circular Motion
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Take note!
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Since objects experiencing uniform circular motion are moving
in a circular path, not only is their direction changing but so it
their velocity. As a result, they are accelerating.
Centripetal Acceleration
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For example, consider an object as it moves from point P
to point Q as shown. If its velocity changes from Vi to Vf
then:
∆V = Vf – Vi
Using triangle congruencies and the equations
V = ∆d/∆t and a = ∆v/t then we can show:
ac = v2/r
Centripetal Acceleration
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Take note!
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Since Vi and Vf are perpendicular
to the radii of the circle, the
acceleration vector points directly
toward the centre of the circle.
Acceleration that is directed
toward the centre of a circular
path is called centripetal
acceleration (ac)
(note)
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Uniform Circular Motion
Occurs when an object moves in a circle and its speed is
constant
Since direction changes the object experiences centripetal
acceleration
Note:
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Centripetal acceleration is always directed toward the centre of the
circle
Centripetal Acceleration
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Practice:
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1. A child rides a carousel with a radius of 5.1 m that rotates
with a constant speed of 2.2 m/s. Calculate the magnitude of
the centripetal acceleration of the child.
Centripetal Acceleration
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Sometimes you may not know the speed of an object
moving with uniform circular motion.
However, you may be able to measure the time it takes
for the object to move once around the circle, or the
period (T)
If the object is moving too quickly, you would measure
the number of revolutions per unit time, or the
frequency (f). Recall:
f = 1/T
In each case, the equation for centripetal acceleration
would become:
(note)
Centripetal Acceleration (more)
Practice
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2. A salad spinner with a radius of 9.7 cm rotates
clockwise with a frequency of 12 Hz. At a given instant, a
piece of lettuce is moving in the westward direction.
Determine the magnitude and direction of the centripetal
acceleration of the lettuce in the spinner at the moment
shown (because, doesn’t everybody wonder how fast their lettuce is
accelerating when making a salad? )
Practice
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3. The centripetal acceleration at the end of a fan blade is
1750 m/s2. The distance between the tip of the fan blade
and the centre is 12.0 cm. Calculate the frequency and
the period of rotation of the fan.
Centripetal Force
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According to Newton’s laws of motion, an object will
accelerate only if a force is exerted on it.
Since an object moving with uniform circular motion is
always accelerating, there must always be a force exerted
on it in the same direction as the acceleration, as shown:
Centripetal Force
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Since the force causing a centripetal acceleration is always
pointing toward the centre of the circular path, it is called
a centripetal force (Fc)
Without such a force, objects would not be able to move
in a circular path
Centripetal Force
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Using Newton’s second law and ac= v2/r the formula for
Fc is:
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**think of Fc as Fnet when dealing with circular motion
(note)
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Centripetal Force
Net force that causes centripetal acceleration (Fc = Fnet)
Centripetal Force
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Take note:
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A centripetal force can be supplied by any type of force
For example, gravity provides the centripetal force that keep
the Moon on a roughly circular path around Earth, friction
provides a centripetal force that causes a car to move in a
circular path on a flat road, and the tension in a string tied to a
ball will cause the ball to move in a circular path when you
twirl it around.
Practice
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4. Suppose an astronaut in deep space twirls a yo-yo on a
string.
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A) what type of force causes the yo-yo to travel in a circle?
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B) What will happen if the string breaks?
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(tension)
(the yo-yp will move along a straight line, obeying Newton’s first law –
objects in motion tend to stay in motion)
5. A car with a mass of 2200 kg is rounding a curve on a
level road. If the radius of the curvature of the road is 52
m and the coefficient of friction between the tires and the
road is 0.70, what is the maximum speed at which the car
can make the curve without skidding off the road?
Practice
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6. You are playing with a yo-yo with a mass of 225
g. The full length of the string is 1.2m.
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A) calculate the minimum speed at which you can swing
the yo-yo while keeping it on a circular path (hint: at
the top of the swing Ft= 0)
B) at the speed just determined, what is the tension in
the string at the bottom of the swing.
7. A roller coaster car is at the lowest point on its
circular track. The radius of curvature is 22 m. The
apparent weight of one of the passengers is 3.0
times her true weight (i.e. FN = 3Fg). Determine
the speed of the roller coaster
Centripetal Force & Banked Curves
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Cars and trucks can use friction as a centripetal force.
However, the small amount of friction changes with road
conditions and can become very small when the roads
are icy
As well, friction causes wear and tear on tires and causes
them to wear out faster
For these reasons, the engineers who design highways
where speeds are high and large centripetal forces are
required incorporate another source of centripetal force
– banked curves
Practice
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8. What angle of banking would allow a vehicle to move
around a curve with a radius of curvature “r” at a speed
“v”, without needing any friction to supply part of the
centripetal force? (In this case you must resolve FN so
that one of the components is directed inward.
Centripetal Force & Banked Curves
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Take Note:
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When an airplane is flying straight and horizontally, the wings
create a life force (L) that keeps the airplane in the air
However, when an airplane need to change directions it must
tilt or bank in order to generate a centripetal force
The centripetal force created is a component of the lift force,
as shown:
Artificial Gravity
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On Earth, the gravity we experience is mainly due to
Earth itself because of its large mass and the fact that we
are on it
However, there is no device that can make or change
gravity
So how can we simulate gravity? The answer is simple –
uniform circular motion
Incorporating the principles of uniform circular motion in
technology has led to advance in
many fields, including medicine,
industry, and the space program
Artificial Gravity
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For example, making a spacecraft rotate constantly can
simulate gravity.
And, if the spacecraft rotates at the appropriate
frequency, the simulated gravity can equal Earth’s gravity
As a result, many of the problems faced by astronauts
working and living in space, such as bone loss and muscle
deterioration, could be eliminated (or at least minimized)
Textobook:
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Pg. 119, #6,8
Pg. 124, #3,4
Pg. 130, #6,7