Download Circular Kinematics

Circular
Kinematics
The London Eye
This Ferris Wheel is
along the Thames in
London. The original
Ferris Wheel was
designed and constructed by George
Washington Gale Ferris, Jr. as a landmark
for the 1893 World’s
Columbian Exposition
in Chicago.
How many machines
using circular motion
are needed to keep
the passengers moving in a circle?
Even though you are
flying in pretty much
a straight line on a
passenger plane,
circular motion still
rules.
From the engines
to the navigation
system, wheels, and
even the doors that
let you in the plane.
If there is no circular
motion, you are not
flying!
T
he wheel is at the center of our “modern” society and everything we have revolves around it. In our mechanical world, it is probably the most important
device ever created. Nearly every machine built since the beginning of the Industrial Revolution invokes the principles of the wheel. It’s hard to imagine any mechanized system that would be possible without the wheel or
the ideas of circular motion that we learn by watching the wheel. From tiny watch gears
to automobiles, jet engines and computer disk drives, the principle is the same.
Working with Circular Motion
The Wheel in Civilization
The wheel is so important that it is hard for us to imagine advanced
civilations that did not use it. But it seems there were many. Either
they were more advanced, or the wheel leads to places they did not
wish to travel. Wheels may be a step backward.
Why? Well, it seems that the wheel just does not exist in the natural
(animal) world on earth! Physics is a natural science and seeks to
explain natural motion.
Later on we’ll learn about circular motion of the planets and subatomic particles, but as far as mother nature here on earth, she either
didn’t like wheels or just didn’t need them!
It appears that the creation of the wheel was done by man. Why? If
the natural world doesn’t need wheels, why do men?
The wheel appears to have been a latecomer to “civilized” cultures
and does not appear to have made much of an impact at all on cultures that the western world calls “uncivilized.” Information Storage
due to Circular Motion
Most “stone-age” peoples of the “new world” had little use for the
wheel in any capacity, but it would probably not be accurate to assume that they were any less civilized than “old world” cultures.
Hawaiians among others had a very highly developed society and
were able to travel long distances over the open seas without any
recourse to the wheel.
Walls Constructed
out of stones with
tolerances greater
than modern
technology (based
on wheels) can
achieve.
The Inca (or pre-Inca among others) are famous for having been able
to construct buildings with massive stone blocks (many of which
cannot be moved today) with tolerances that cannot be achieved
with today’s technologies. The ancient pyramids remain a mystery. Since “modern” man cannot conceive of methods to construct them without a wheel of some
type, most theories rely on rolling the stones over circular objects
that functioned like wheels (dragging the blocks over trees, etc.).
Virtually all modern transportation systems are based on the
wheel. All electrical power generation and distribution systems
need a wheel, our information systems are still dominated by the
wheel. Future ones may not be.
So is the wheel a symbol of an advanced civilization, or is it a limiting idea?
Circular Kinematics
Rotational Vectors
The Right Hand Rule
To study circular motion, we will need to have a way to determine
how many times a circle spins (or how many times you traveled
around it), and how fast and in which direction it is spinning. You
have done that in previous classes by using angles measured in degrees. We’ll still do that, but we are also going to learn to measure angles
another way, we are going to use the radius of the circle itself to
measure the angle that the circle turns.
Let’s get started.
If it is rotating clockwise, the rotational vector is pointing AWAY from you.
If it is rotating counter-clockwise, look out,
because it is pointing at you.
Circles, Angles, and Radians
So, if you walk around a circle, how far did you walk?
You’ll probably answer something like “it depends on how big the
circle is.”
But does it?
If you really are trying to mess with the physics teacher, you’ll say
ZERO because you are thinking about linear displacement. And you
would be right, except for the fact that I asked about how far (a distance) and not for your linear displacement.
Since this section is about circular motion, I might also ask you
what your angular displacement was. How much angle did you
walk? That is definitely not zero, unless you walk back around again
(the opposite way).
Angular displacement is a vector, it tells us how much we (or whatever) have rotated since the start of the problem. We usually use
radians to measure angles when we are talking displacement, so it’s
time to learn about radians. Work through the next animation to
learn what a radian is.
Angular Displacement
A jogger is running on a circular track with a
radius of 50.0m.
What is her angular displacement when she
runs 20.0m?
How far has she run when her angular displacement is 5.0 radians?
What is her angular displacement when she
has run 1.0 km? (Answer in both radians and
revolutions.)
An automobile tire is guaranteed for 40,000
miles.
If the diameter of the tire is about 3.0 feet,
how many revolutions will the tire make in it’s
lifetime?
What will it’s angular displacement in radians
be (assuming that it doesn’t go in reverse).
The Cartesian coordinates of a point are
(-2.5, 5.9). What are the polar coordinates?
In Cartesian coordinates, the equation of a
circle is x2 + y2 = r2. What is the equation using polar coordinates?
The radius of a circle fits around the circle
exactly 2*PI times.
Create a conversion factor to change from
radians to degrees.
Create a conversion factor to change from
degrees to radians.
The pizza delivery person brings a 24 inch
(diameter) pizza to your school for the radian
physics party.
How many wedge shaped pieces can be cut
with a crust length equal to the radius of the
pizza.
What is the crust length of the remaining
piece?
Small Angle Approximation
So, the idea is really pretty simple. A radian is just the distance
you travel along the curve (arc length) divided by the radius of the
curve. When angles are very small, the difference
between the arc length and the chord length
(b) becomes so small that we can ignore the
difference. In other words, if the difference is smaller
than we can measure, we can assume that
the two are equal.
Definition of a Radian
Arc length (s) divided by
the radius (r).
One radian is the angle that a radius makes along the curve (arc).
One radian is the angle that a radius makes
along the curve (arc).
This approximation makes a lot of scientific
formulas much easier to deal with. It may
seem like cheating, but it’s not cheating
if you cannot calculate the difference, and
this situation happens in the real world often
enough to use small angle approximations
when necessary.
Small Angles
So if I ask you how far (in radians) you traveled by walking around
a circle, it does not depend on how big the circle is. The angular displacement is the same for all circles. To put it simply, you walked
all the way around. The answer is the same for both circles; one
revolution, 360 degrees, or 2π radians.
On the other hand, if I ask about linear displacement, then you obviously travel farther around circles that have a bigger radius.
A submarine sailor using
his persicope sees a ship
on the surface. From past
experience, he knows the
ship is 15m tall. The periscope contains divisions
that tell the submariner
that the ship’s height
takes up 2.0 degrees of
arc.
One more thing to note. A radian is a length divided by a length. The
basic dimensions cancel which means that a radian has no basic dimensions. It is just a number.
How far away is the ship?
Linear and Angular Variables
Small angle approximations are usually used
when the sine of an angle is pretty much the
same as the angle itself.
If you measure an angle to be 1.1 radians (2
significant figures), can you use small angle
approximations for problems that use this angle?
If you measure an angle to be 0.010 radians,
can you use small angle approximations for
calculations that use this measurement?
... a radian has no basic dimensions. It is just
a number.
Now, since s is a length, we can use the formula for a radian to help
us make the kinematic equations for circular motion. Before we do that, let’s make some letters/symbols that represent
angular displacement, angular velocity (how fast is it spinning),
and angular acceleration (is it spinning faster or slower).
Symbols for Angular Displacement
Velocity and Acceleration
The three letters we’ll use are theta-θ, omega-ω , and alpha-α.
Let’s start with the first one. The greek letter theta-θ is normally
used to indicate an angle. Our angles will usually be in radians, so
on the right I have the formula for radian measure (s / r). The letter
s is a linear displacement. I know what you’re going to say. I used
d or x for linear displacement in the past, so why am I using s now? It’s simple! S is a curvy letter, and we are dealing with curves!!! Got
it?
Now that you know how s is related to θ, you can use the formulas for angular velocity (omega), and angular acceleration (alpha)
to create the relationships between linear and angular displacement,
velocity, and acceleration. Here they are.
Formulas to convert from Linear Kinematics
to Circular Kinematics
Sooo then. All you gotta do to convert from angly stuff to straight
stuff is slap it with a radius. (Uh, I mean multiply it.) That should
make sense. The radius is the straight thing that creates a curvy
one. Each of the properties on the left above is a linear one. It is also
called tangential because they point in a direction that is tangent
to the circle. So when you hear the words tangential velocity, tangential acceleration, and tangential displacement, you know that
we are talking about vectors that are parallel (or anti-parallel) to the
direction of the motion AT THAT INSTANT in time!
One more thing here. These angular properties are vectors! Remember those? They have magnitude and direction. The magnitude tells
us how fast it’s spinning, how much it did spin and if it’s speeding
up or slowing down. The direction tell us which way it’s spinning or
which way it’s changing.
We will still use arrows to represent vectors, but these arrows are going to seem a little bit weird at first because they are not in the plane
of the thing that is spinning. Instead, they point along the axis of
rotation - kind of just out into space! In order to know which way the
vector is pointing, you will need to use the right hand rule. (Watch
the rotaional vector video in the sidebar.)
Angular Velocity & Frequency
How fast does it go around the circle? Well, how fast does it go aroung the circle?
Some gifted people can run around a 400m
track in one minute. And, they can do it over
and over (at least 4 times in a row).
That means they are running the track with
an angular frequency of ONE rpm - one
rotation per minute.
What do I mean when I say frequency? Frequency means HOW FREQUENTLY, HOW
MANY TIMES DOES IT GO ROUNDY ROUNDY;
completely around! Each time it goes all the
way around is called a REVOLUTION. Now, think carefully. If it goes around 1 time
every minute, then the frequency is 1 rpm
or one rotation per minute. But wait, a rotation is 2*PI radians, so the frequency tells
us how fast the object is rotating. In other words, it is an angular velocity!
It may not be in radians, but it still lets us
know how fast is it spinning!
Frequency & Period
If you run around a 400m track 4 times in
4 minutes, how long does it take you to go
around once?
The answer is obviously 1 minute. The PERIOD of time that it takes to complete one
cycle, or one revolution is called the period
(because it’s the period of time.) See if you
can figure out how the period is related to
the frequency.
Revolutions, Cycles, Hz
A cycle is a completion of a repeating pattern.
The pattern in circular motion is one complete revolution. So in circular motion, one
revolution is one cycle. (There are other
kinds of repeating motions that are not
circular.)
65 cycles means 65 revolutions in circular
motion.
One cycle per second has a special name - it
is known as one Hertz (or 1 Hz).
Our electrical power grid revolves 60 times
per second - 60 cycles per second - 60 Hz.
Units and Dimensions
Angles can be measured a lot of different
ways. You probably learned about degrees
first. Then you learned rotations (how many
times does it spin) and now you know about
radians. Angular Kinematic Equations
And now, for the grand unveiling of the new kinematic equations for
circular motion. . . We’ll do this by just substituting the new letters
into the old kinematics equations.
All of these usually show up as a way to describe angular velocity (frequency).
Common Angular Displacement Units
Type of
Unit
Basic
Dimensions
SI (mks)
radians/rads
revolutions
revs
Cycles
There are
no Basic
Dimensions
for angles.
It is just a
number!
Common Angular Velocity Units
Type of
Unit
Units
SI (mks)
rads / sec
T-1
rpm
rev / min
T
Hz
cycles / sec
T-1
Basic
Dimensions
-1
Common Angular Acceleration Units
Type of
Unit
Units
SI (mks)
rads / sec2
Basic
Dimensions
T-2
Angular Velocity
A spinning bicycle wheel is rotating counterclockwise (CCW) when you look at it.
Which way is the angular velocity vector pointing? Up, down, left, right, toward
you, away from you, or something else?
Linear Kinematics formulas on the Left
Angular Kinematics formulas on the Right
Wow, no big surprise there huh. All we did was change the letters
to greek. Hey!
It’s all greek to me.
None of the basic ideas have changed. If you understood linear kinematics, you should be able to do this too. Everything we said about
averages and instantaneous values for linear kinematics is also true
for circular kinematics.
And now we can do problems that ask us to convert back and forth
from circular to linear motion. Wow, good fun!
So how do objects move in circles? Is there a special kind of circular
force or something? And if it is moving in a circle, how do you keep
it moving that way?
Do all points on the spoke of the wheel have
the same linear velocity?
Look back at the Eye of London picture on the opening page of this
chapter. The Eye is a ferris wheel that has passengers moving in a
circular path. What force keeps them moving in a circular path?
Do all points on the spoke of the
wheel have the same angular velocity?
Let’s look at some things that move in circular paths.
Your car engine’s crankshaft is spinning at
700 rpm as shown on the tachometer.
What is the angular velocity of the engine in
radians per second?
What is the frequency of the engine in Hz?
A satellite in a circular orbit has a period of
10 hours.
What is the satellite’s frequency in revolutions
per day? What about in Hz?
A car makes 2.5 laps around a circular track
with a radius of 85m in 3.0 min.
What is the car’s average angular speed?
What is the car’s tangential speed?
The planets in our solar system move
in nearly circular paths around the sun.
What force is causing them to move
in a circle. The speed of the planets
doesn’t change much, but their direction does.
The same is true of the moon moving
around the earth in a nearly circular
path. Earth’s artifical satellites like the
space station and other communications satellites also move in circular paths around the earth. What kind of force causes this circular
motion?
These wind turbines have blades that spin in circles. What type of
force would keep the outer ends of these blades moving in a circle.
Which way is the force directed?
Angular Acceleration
A merry-go-round with a diameter of 11 meters rotates once every 8.0 seconds.
If it takes 3.5 seconds to get up to speed
(from rest), what is the average angular acceleration during startup.
After the ride is complete, the merry-go-round
slows to a stop. If it rotates 3/4 of a revolution while stopping, what was its average
angular acceleration during this time.
A potter’s wheel moves from
rest to an angular speed of
0.20 rev/s in 30 s.
Find its angular acceleration in radians per
second per second.
If the diameter of the wheel is 0.55 m, what
is the magnitude of the tangential acceleration
of a lump of clay on the edge of the wheel?
These motorcycles are
making a circular turn.
How do they do it?
A car traveling at 45 mph slows to 30 mph in
15 seconds.
If the diameter of the wheels is 3.0 feet, what
is the average angular acceleration in radians
per sec2?
And what about these
horses turning?
Rotational Kinematics
A dentist’s drill starts from rest. After 3.20 s
of constant angular acceleration, it turns at a
rate of 2.51 × 104 rev/min.
Find the drill’s angular acceleration.
Determine the angle (in radians) through
which the drill rotates during this period.
A centrifuge in a medical laboratory rotates
at an angular speed of 3,600 rev/min. When
switched off, it rotates through 50.0 revolutions before coming to rest.
Or the riders in this roller-coaster?
Find the constant angular acceleration of
the centrifuge. (In radians per second per
second.)
A machine part rotates at an angular speed
of 0.60 rad/s; its speed is then increased to
2.2 rad/s at an angular acceleration of 0.70
rad/s2.
Find the angle through which the part rotates
before reaching this final speed.
The diameters of the main rotor and tail rotor
of a single-engine helicopter are 7.60 m and
1.02 m, respectively. The respective rotational speeds are 450 rev/min and 4,138 rev/
min.
Calculate the speeds of the tips of both rotors.
Compare these speeds with the speed of
sound, 343 m/s.
Or this one?
The tub of a washer goes into its spin-dry
cycle, starting from rest and reaching an
angular speed of 5.0 rev/s in 8.0 s. At this
point, the person doing the laundry opens the
lid, and a safety switch turns off the washer.
The tub slows to rest in 12.0 s.
Through how many revolutions does the tub
turn during the entire 20-s interval? Assume
constant angular acceleration while it is starting and stopping.
A coin with a diameter of 2.40 cm is dropped
on edge onto a horizontal surface. The coin
starts out with an initial angular speed of
18.0 rad/s and rolls in a straight line without
slipping.
If the rotation slows with an angular acceleration of magnitude 1.90 rad/s2, how far does
the coin roll before coming to rest?
How many seconds does this take?
Add Gear Ratio Problems
Here is an interesting
situation. The truck
was driving over a
circular shaped sand
dune, but it doesn’t
move in a circle.
What happened? Is
there no circular force
for trucks in the sand?
Or was the motion of
the truck too much for
the force?
two kinds of circular accelerations and forces
Centripetal Acceleration
BIG TIME Circular Motion
Way back at the beginning, we learned that objects have inertia, and
that their inertia leads to them moving in straight paths at constant
velocity unless some outside force gets involved.
It seems pretty clear that there must be some kind of outside force
acting on objects that move in circles. Forces cause acceleration. .
. so . . . to put it as easy as I can, things don’t move in circles unless
there is acceleration going on. Why? Because velocity is a vector. It has magnitude and direction. I’ve said that a million times now, and I’ll probably say it a
million more. What is going on here? What is forcing these
cars and bike riders to move in a circle? If you are moving in one direction, you’ll keep moving that way unless you turn. Pretty simple huh? Well if you turn, you just accelerated, even if you didn’t change your speed. Things that are moving in a circle (including all the molecules in a
wheel that is spinning), are constantly turning which means that they
must be constantly accelerating. The question is which way? Le Rotor
So the acceleration - centripetal acceleration - is toward the center,
toward the point around which the things rotate. It’s like they are
constantly trying to get to the center but never make it there! Don’t
forget that. There is no such thing as a centrifugal acceleration (or a centrifugal
force) that makes something go in a circle. There is no such thing as a centrifugal acceleration (or a centrifugal force) that makes something go in a circle.
The formula for centripetal acceleration can be written 2 ways. One
involves the linear velocity (tangent to the path) and the other involves the angular velocity. The derivation of the formula involves
some basic geometry and similar triangles; see the sidebar discussion.
Oh my gosh! These people stick to the walls
too, just like the last video. Where can you
buy these kinds of walls. How do they make
them?
Uniform Circular Motion
Physics books like to talk about something
called uniform circular motion.
It sounds like its a mystery or something
when it’s said like that.
So let’s keep it simple.
A particle moving in uniform circular motion moves in a circle at a constant angular velocity and a constant radius.
This means that an object that is spinning at
a constant angular velocity has all of its
molecules moving in uniform circular motion (the same angular speed).
Uniform circular motion simply means
that everything is moving is a circule together at a constant angular velocity! Centripetal Acceleration
Remember, centripetal acceleration is a vector. The direction of
the vector is constantly changing in circular motion but it always
points to the center of the circle (or of the rotation).
This idea can be a little bit
tough to grasp without lots
of thought and practice
working the problems. I introduced the formula above
for centripetal acceleration,
but sometimes a less mathematical approach works
better to help you get the
idea.
Click on the picture of the
circle and triangle here for a graphical explanation of centripetal
acceleration and forces.
Not too hard.
Video explanations of Circular Motion
There’s lots and lots of good stuff to think
about in this video.
Of course, that means you have to watch it
and think about it.
Centripetal Acceleration
In uniform circular motion, there is
(a) a constant velocity.
(b) a constant angular velocity.
(c) zero acceleration.
(d) a non-zero tangential acceleration.
Centripetal Forces
Acceleration is caused by force. You have to force things to revolve
in a circle, they don’t seem to do it willingly.
Newton’s second law says that
net force is the product of mass
times acceleration. If the centripetal force on a particle in uniform circular motion is increased,
(a) the tangential speed will not change.
(b) the tangential speed will decrease.
(c) the radius of the path will increase.
(d) the radius of the path will decrease.
In a washing machine, the spin cycle is used
to remove water from the clothes.
Explain how this works!
The ball on the rope is held in circular motion by the tension in the
rope. It is this rope that is causing
the circular motion.
Circular motion is caused by
real forces that you can identify,
like a rope tied to a rock that you
swing in circles, or the force of
gravity causing the moon to go
around the earth. These are real
forces (tension and gravity) and
they are causing circular motion.
What I am trying to say is that circular motion is caused by the
NET force; by one or more real forces. It is not a new magical
force. Since it is a NET force, we can write an equation for it by using Newton’s second law. Here it is.
In the picture we have two jars with fishing bobs (floating) in jars partially filled with
water. We are going to start the arms of the
machine spinning.
Which way will the floats move? Will they
move? Does it matter which way we spin the
arms?
A psycho physics teacher swings a cup of
water around on a plate attached to a rope as
shown in this video (link). Assume the rope
is 0.50 m long.
If he swings the ball around 2 times per
second, what is the centripetal acceleration of
the ball?
What if the rope was only 0.25 meters
long. What would the centripetal acceleration
be then?
OK, back to the rope being 0.50 meters
long. If the nutso teacher swings the rope
twice as fast (4.0 times per second) what is
the centripetal acceleration?
Space stations could be
designed to
rotate (as
a ring or a
cylinder) so
that they
create a
centripetal
force pushing inward
on objects
on their
inner walls. This force simulates the earth’s
surface pushing up on your feet to counteract
the force of gravity. Suppose we design such a space station that
is 5.0 mi in diameter.
Centripetal Force
We call the net force that makes something move in a circle a
centripetal force. I’ll say this again, centripetal force is a NET
FORCE! It is not a new magical circular force!
. . . centripetal force is a NET FORCE! It is not
a new magical circular force!
Remember that centripetal means “center-seeking.” So a centripetal force will push or pull objects toward the center and the result
will be circular motion. It’s worth repeating again that circular motion is not caused by an
outward force that some people call centrifugal force. In fact, there
is no such thing as a centrifugal force that pushes you to the outside
of a circle. Things that appear to be forced to the outside of a circle
are just moving in a straight line like they’re supposed to. When you
feel pushed to the outside of a turn in your car or on a carnival ride,
you are moving straight. Everything else is experiencing centripetal acceleration toward the center which makes it appear like YOU
are being pushed to the outside, but it just ain’t so!
If you want to make something that is moving with a velocity v
move in a circle, you will have to push or pull on it with a force that
is equal to the square of that velocity divided by the radius of the
circle you want to have. If you don’t, it will not move in circular
motion.
Some Applications of
Centripetal Force
There are a lot of classic problems that test your understanding of
centripetal forces. I have grouped them into a few basic categories
below.
CENTRIPETAL FORCE IS A NET FORCE
Centripetal force is not a force that belongs on your free body diagram.
Centripetal force is not a force that belongs on
your free body diagram.
Why? Because it is the NET force. It is the force that is the result of
the real forces that should be on your free body diagram. Watch this
ki`i training show.
What angular speed must such a cylinder
have so that the centripetal acceleration at
its surface equals the free-fall acceleration on
Earth?
A bug sits on the rim of a 10-in.-diameter
disk that moves from rest to an angular
speed of 78 rev/min in 3.0 s?
What is the tangential acceleration of the bug.
When the disk is at its final speed, what is the
tangential velocity of the bug?
One second after the bug starts from rest,
what are its tangential acceleration, centripetal acceleration, and total acceleration?
Centripetal Force
For the car in this
picture to move in
a circle, there must
be a force pushing it
inward - the centripetal force.
Where does the centripetal force come from
that allows the car to turn?
If the seats are slippery, the passengers feel
like they are being pushed to the outside of
the turn. Why do they feel this? Is there a
force that is pushing them outward?
If the car moving at 83 km/hr goes around a
turn with a 40.0 meter radius of curvature,
what is the car’s centripetal acceleration?
Once again, the centripetal force is the result of the real forces. There
is no magical “circular force.” Centrifugal force is an illusion. You
only think you are being forced to the outside of a circle because
you think the circle is natural. It is not. It requires a force to go in a
circle, not to leave it.
Centrifugal force is an illusion. You only think
you are being forced to the outside of a circle
because you think the circle is natural. It is
not. It requires a force to go in a circle, not to
leave it.
If the coeffecient of static friction between the
tires and the road is 0.65, will the car make it
through the turn without slipping?
In the previous problem, there had to be
friction between the car and the tires in order
to push the car inward toward the center of
the circle. Sharp curves on highways and
racetracks are often designed with banked
turns (inclined) so that there will be a source
of centripetal force even without friction.
Draw a free body diagram of a car in a banked
turn on a frictionless surface (look at the front
or the back of the car and have the bank going up to the left or up to the right).
Assuming that the Earth can be approximated
as sphere that rotates about an axis connecting the north and south poles, and that it has
a radius of 6,384 km.
Find the centripetal accelerations of (a) a
point on the equator of Earth and (b) the
North Pole, due to the rotation of Earth about
its axis. (Answer in m/s2 .)
What is the weight of a 65.0 kg person standing on the equator?
What is the weight of a 65.0 kg person standing on the north pole?
A 55.0-kg ice-skater is moving at 4.00 m/s
when she grabs the loose end of a rope, the
opposite end of which is tied to a pole. She
then moves in a circle of radius 0.800 m
around the pole.
Determine the force exerted by the horizontal
rope on her arms.
Compare this force with her weight.
HORIZONTAL CIRCLES AND
FREE BODY DIAGRAMS
The axes on your free body diagrams should be setup so that the
NET CENTRIPETAL FORCE is in the same plane. Just because
there is a slope does not mean you should rotate the axes. You should
only rotate the axes if the motion is not in the same plane as one of
your coordinate axes.
A 50.0-kg child stands at the rim of a merrygo-round of radius 2.00 m, rotating with an
angular speed of 3.00 rad/s.
What is the child’s centripetal acceleration?
What is the minimum force between her feet
and the floor of the carousel that is required
to keep her in the circular path?
What minimum coefficient of static friction is
required? Is the answer you found reasonable? In other words, is she likely to stay on
the merry-go-round?
An engineer wishes to design a curved exit
ramp for a toll road in such a way that a
car will not have to rely on friction to round
the curve without skidding. He does so by
banking the road in such a way that the force
causing the centripetal acceleration will be
supplied by the component of the normal
force toward the center of the circular path.
VERTICAL CIRCLES
INSIDE AND OUSTIDE LOOPS
(a) Show that, for a given speed v and a radius r, the curve must be banked at the angle
θ such that tan θ = v2/rg.
(b) Find the angle at which the curve should
be banked if a typical car rounds it at a 50.0m radius and a speed of 13.4 m/s.
An air puck of mass 0.25 kg is tied to a string
and allowed to revolve in a circle of radius
1.0 m on a frictionless horizontal table. The
other end of the string passes through a hole
in the center of the table, and a mass of 1.0
kg is tied to it. The suspended mass remains
in equilibrium while the puck on the tabletop
revolves.
(a) What is the tension in the string?
(b) What is the horizontal force acting on the
puck?
(c) What is the speed of the puck?
Tarzan (m = 85 kg) tries to cross a river by
swinging from a 10-m-long vine. His speed at
the bottom of the swing (as he just clears the
water) is 8.0 m/s. Tarzan doesn’t know that
the vine has a breaking strength of 1,000 N.
Does he make it safely across the river?
Justify your answer.
These types of circles can cause the object to fall (inside loop) or
fly off the circle (outside loop) if there is not enough energy (inside
loop) or if there is too much (outside loop).
The critical energy is the same for both types of loops, but the result
is not.
TOO STRONG FOR THE
ALLOWABLE CENTRIPETAL FORCE
Everything that moves in a circle has a net force that is pulling toward the center.
Since the NET force must account for the speed of the object and the
radius of the circle, moving too fast will cause the object to break
free of the force and travel outside the circle. Not only that, moving
in too small of a circle will also cause the object to break free of the
force and travel outside the circle.
On the other hand, moving too
slow will cause the circle to
shrink or break down altogether.
The rope is creating the centripetal force. If you try to make
the ball go in a circle too small
or too fast, the rope will not be
able to hold onto it.
Here we have the boy swinging a
ball on a rope again. The rope is
under tension. It is pulling the ball
toward the center of the circle.
The faster he swings the ball,
the stronger the force must be to
keep it in a circular path.
Eventually, if he swings it too fast, the rope will not be able to hold
the ball and it will break allowing the ball to go sailing off.
As the rope rotates, each part of the rope pulls on the parts next to
it causing it to rotate in circular motion. Since all parts of the rope
have the same angular speed, the parts that are closer to the ball
experience a greater tension than the parts closer to the boy’s hand. Here’s another example of what
happens when the available centripetal force is not strong enough
to ensure circular motion.
If the vehicle has a speed of 20.0 m/s at point
A, what is the force of the track on the vehicle
at this point?
What is the maximum speed the vehicle can
have at point B in order for gravity to hold it
on the track?
One method of pitching a softball is called
the “windmill” delivery method, in which the
pitcher’s arm rotates through approximately
360° in a vertical plane before the 198-gram
ball is released at the lowest point of the
circular motion. An experienced pitcher can
throw a ball with a speed of 98.0 mi/h. Assume that the angular acceleration is uniform
throughout the pitching motion, and take the
distance between the softball and the shoulder joint to be 74.2 cm.
(a) Determine the angular speed of the arm in
rev/s at the instant of release.
(b) Find the value of the angular acceleration
in rev/s2 and the radial and tangential acceleration of the ball just before it is released.
This chimney was blasted at the
base. It began to rotate toward
the ground.
(c) Determine the force exerted on the ball by
the pitcher’s hand (both radial and tangential
components) just before it is released.
Each brick of the chimney was
bonded to the one next to it.
The bricks farther away from the
rotation point will have to bond
with a greater force in order to
keep the chimney rotating in one
piece.
A rollercoaster
vehicle has
a mass
of 500 kg
when fully
loaded
with passengers.
In a popular amusement park ride, a rotating cylinder of radius 3.00 m is set in rotation
at an angular speed of 5.00 rad/s. The floor
then drops away, leaving the riders suspended against the wall in a vertical position.
Demolition of a smokestack in
Kentucky.
Eventually, the increasing angular speed will cause some of the bonds to break because they can no
longer provide the force required to maintain circular motion.
Centripetal and Tangential
Forces
Before ending this chapter, we have to talk about circular motion
forces a bit more.
There are really two types of forces that we will be dealing with
whenever we talk about circular motion. The centripetal force we
now know is the one that pushes or pulls toward the center of the
circle. The acceleration is called centripetal acceleration.
Centripetal forces are always perpendicular to the motion of the object which means they cannot cause the object to speed up or slow
down. (No work is done by forces that are perpendicular to the dis-
What minimum coefficient of friction between
a rider’s clothing and the wall is needed to
keep the rider from slipping?
A massless spring of constant k
= 78.4 N/m is fixed on the left
side of a level track. A block of
mass m = 0.50 kg is pressed
against the spring and compresses it a distance d.
The block (initially at rest) is then released
and travels toward a circular loop-the-loop
of radius R = 1.5 m. The entire track and the
loop-the-loop are frictionless, except for the
section of track between points A and B.
Given that the coefficient of kinetic friction
between the block and the track along AB is
μk= 0.30, and that the length of AB is 2.5 m,
determine the minimum compression d of the
spring that enables the block to just make it
through the loop-the-loop at point C.
Tangential and Centripetal Acceleration
placement. If there is no work done, then the kinetic energy of the
object does not change.) That’s why we can have a formula for centripetal acceleration that is related to the magnitude of the velocity;
because velocity doesn’t change when only centripetal acceleration
is occuring.
But clearly we can cause objects to move faster and slower in circular motion. The force that does that is connected to something we
call torque. Torque will cause the object moving in circular motion
to accelerate in a direction that is tangent (parallel) to the motion; it
will cause circular moving objects to speed up and slow down.
This softball pitcher is causing both centripetal and tangential acceleration of the ball.
One causes the ball to speed up (tangential)
and the other causes it to stay in a circle
(centripetal).
Torque is the subject of our next chapter - circular dynamics, so we
won’t be dealing with it much here, but you should still understand
that these forces cause tangential acceleration and not centripetal
acceleration. We don’t deal with torque here, because torque is essentially the application of Newton’s laws of motion to rotations.
Problems
1. Assume that
the Earth’s orbit
around the sun is
circular.
tive forces, determine (a) the spring constant
and (b) the speed of the projectile as it moves
through the equilibrium position of the spring
(where x = 0), as shown in the right picture.
a) What is the
orbital distance that
the earth travels in
24 hours?
6. A rotating wheel requires 3.00 s to rotate 37.0
revolutions. Its angular velocity at the end of
the 3.00-s interval is 98.0 rad/s. What is the
constant angular acceleration of the wheel?
b) How many
times does the earth
rotate on its axis in
24 hours (answer
in revolutions and
radians)
7. A cylinder 10.0
mi long and 5.0
mi in diameter is
designed to rotate
about its long central axis in space
as a space station.
The people will
live on the inside
of the cylinder
walls. How fast
does the cylinder need to rotate in order to
simulate the gravity felt on the surface of the
earth?
2. On a particular clock, the lengths of the hands
are:
Second hand:
Minute hand:
Hour hand:
18 cm
22 cm
8.0 cm
How far does the tip of each hand travel in a 20
minute period?
3. Your car’s engine (the crankshaft) is spinning
at 700 rpm as shown on your tachometer. The
tires on your car have a diameter of 0.75m.
Your speedometer indicates that you are traveling at a speed of 10.0 mph.
a) What is the angular velocity of the engine
in radians per second? b) What is the angular velocity of the tires in radians per second?
c) What is the mechanical advantage of the gear
you are in (the ratio of the engine speed to the
tire speed)?
4. The tub of a washer goes into it’s spin-dry
cycle, starting from rest and reaching an angular
speed of 5.0 rev/s in 8.0 s. At this point, the person doing the laundry opens the lid, and a safety
switch turns off the washer. The tub slows to
rest in 12.0 s. Through how many revolutions
does the tub turn during the entire 20-s interval?
Assume constant angular acceleration while it is
starting and stopping.
5. A coin with a diameter of 2.40 cm is dropped on
edge onto a horizontal surface. The coin starts
out with an initial angular speed of 18.0 rad/s
and rolls in a straight line without slipping. If
the rotation slows with an angular acceleration
of magnitude 1.90 rad/s2, how far does the coin
roll before coming to rest?Neglecting all resis-
8. The moon revolves around Earth in 29.5 days
in a nearly circular orbit. The radius of the
orbit is 3.8 x 105 km. What is the acceleration
of the moon?
9. (a) What is the tangential acceleration of a
bug on the rim of a 10-in.-diameter disk if the
disk moves from rest to an angular speed of 78
rev/min in 3.0 s? (b) When the disk is at its
final speed, what is the tangential velocity of
the bug? (c) One second after the bug starts
from rest, what are its tangential acceleration,
centripetal acceleration, and total acceleration?
10. A 50.0-kg child stands at the rim of a merrygo-round of radius 2.00 m, rotating with an
angular speed of 3.00 rad/s. (a) What is the
child’s centripetal acceleration? (b) What is
the minimum force between her feet and the
floor of the carousel that is required to keep
her in the circular path? (c) What minimum
coefficient of static friction is required? Is the
answer you found reasonable? In other words,
is she likely to stay on the merry-go-round?
11. An engineer wishes to design a curved exit
ramp for a toll road in such a way that a car
will not have to rely on friction to round the
curve without skidding. He does so by banking the road in such a way that the force
causing the centripetal acceleration will be
supplied by the component of the normal force
toward the center of the circular path. (a) Show
that, for a given speed v and a radius r, the curve
must be banked at the angle θ such that tan θ
= v2/rg. (b) Find the angle at which the curve
should be banked if a typical car rounds it at a
50.0-m radius and a speed of 13.4 m/s.
12. An air puck of
mass 0.25 kg is
tied to a string
and allowed to
revolve in a circle
of radius 1.0 m
on a frictionless
horizontal table.
The other end of
the string passes through a hole in the center of
the table, and a mass of 1.0 kg is tied to it. The
suspended mass remains in equilibrium while
the puck on the tabletop revolves. (a) What is
the tension in the string? (b) What is the horizontal force acting on the puck? (c) What is the
speed of the puck?
13. A roller-coaster vehicle has a mass of 500 kg
when fully loaded with passengers as shown
in the picture. (a) If the vehicle has a speed
of 20.0 m/s at point A, what is the force of the
track on the vehicle at this point? (b) What is
the maximum speed the vehicle can have at
point B in order for gravity to hold it on the
track?
14. One method of pitching a softball is called
the “windmill” delivery method, in which the
pitcher’s arm rotates through approximately
360° in a vertical plane before the 198-gram
ball is released at the lowest point of the circular motion. An experienced pitcher can throw a
ball with a speed of 98.0 mi/h. Assume that the
angular acceleration is uniform throughout the
pitching motion, and take the distance between
the softball and the shoulder joint to be 74.2 cm.
(a) Determine the angular speed of the arm in
rev/s at the instant of release. (b) Find the
value of the angular acceleration in rev/s2 and
the radial and tangential acceleration of the ball
just before it is released. (c) Determine the
force exerted on the ball by the pitcher’s hand
(both radial and tangential components) just
before it is released.
15. In a popular amusement park
ride, a rotating cylinder of radius 3.00 m is set in rotation at an
angular speed of 5.00 rad/s. The
floor then drops away, leaving
the riders suspended against the
wall in a vertical position. What
minimum coefficient of friction between a rider’s clothing
and the wall is needed to keep
the rider from slipping? (Hint:
Recall that the magnitude of the maximum force
of static friction is equal to μn, where n is the
normal force—in this case, the force causing the
centripetal acceleration.)
16. A massless spring of constant k = 78.4 N/m is
fixed on the left side of a level track. A block of
mass m = 0.50 kg is pressed against the spring
and compresses it a distance d, as shown below.
The block (initially at rest) is then released and
travels toward a circular loop-the-loop of radius
R = 1.5 m. The entire track and the loop-theloop are frictionless, except for the section of
track between points A and B. Given that the
coefficient of kinetic friction between the block
and the track along AB is μk = 0.30, and that the
length of AB is 2.5 m, determine the minimum
compression d of the spring that enables the
block to just make it through the loop-the-loop
at point C. (Hint: The force exerted by the track
on the block will be zero if the block barely
makes it through the loop-the-loop.)