Download Circular Motion and Gravitation

Circular Motion and Gravitation
Let's consider things that go round and round!
Rotation and Revolution:
These words have very specific meanings in physics. The difference is simple. Something that rotates
spins about an axis that passes through the object. Merry-go-rounds, planets and car wheels rotate.
Things that revolve spin about an axis that falls outside of the object. Maypole dancers, cars in a
roundabout, and planets revolve.
Note that I cited planets in both of the examples above. The fact is that an object can both rotate and
revolve, even at the same time. The earth rotates about its own axis once per day and revolves about
the sun once per year.
Units of rotation:
There are several units of rotation, each good for its own purpose. The simplest is the revolution, or
complete turn. Another is the degree, of which there are 360 in a complete turn.
Particularly useful is the radian, a unit of angular measure based upon a
unit circle. If you draw a circle of one unit radius, then go a distance of one
radius along the circumference of the circle you will describe an angle of
one radian. Of course, the formula for circumference is Circ=2πr so 2π
radians is a complete circle or 360 degrees. The cool thing about radians is
that you can get the distance along the circumference simply by multiplying
the radius by the angle in radians.
r
One
Radian
r
Centripetal acceleration and force:
If an object is moving in a circular path it is changing velocity, so therefore it is accelerating. The
speed does not need to change for acceleration to occur. The acceleration is determined by the formula
ac=v2/r
Where ac is the centripetal acceleration, v is the tangential velocity (velocity along the curved path) and
r is the radius of the path.
Of course, to make something accelerate, there has to be an outside force upon it. A force, or
combination of forces, that makes something move in a circular path is called a centripetal force, and is
symbolized by Fc. The familiar second law formula is used to determine this:
ΣFc=mac
Note that Fc is not a separate force, but is the sum of whatever forces are making an object go in a
circle. In a car going around a corner the Fc is provided by the sideways friction force acting upon the
car's tires. In a tetherball, the Fc is provided by the tension in the tether. In the case of a child on a
swing at the lowest point, the Fc is provided by the sum of the tension forces in the ropes and the
gravitational force acting upon the child. Since the second law is in action, it is very useful to draw a
FBD of the situation before attempting to solve it
For an object moving in a circle with no gravity involved (this includes horizontal circles in a
gravitational field) finding ac and Fc is easy. Here's an example:
Let's consider a 3.0kg mass traveling in a circle on a
frictionless tabletop tethered by a cord of length 2.0m. The
velocity of the mass is 7.0 m/s. We'd like to find the centripetal
acceleration and the tension in the cord.
7m/s
2m
ac=v2/r
ac=72/2=24.5m/s2
To find the centripetal force, we first draw a FBD of the mass:
Ft
The only force is that of tension, which provides the centripetal force. Therefore:
ΣFc=mac
Ft=3.0(24.5)=73.5N
How come everyone else calls it “centrifugal” force?
Because they're uneducated, that's why!
“Centripetal means “center-seeking” and that is indeed what a centripetal forces does, make an object
accelerate toward the center of a circle. It's a real force and it has a real effect.
“Centrifugal” means “center-fleeing” and it's an apparent reality only to someone observing it from a
rotating reference frame. If you are inside a car that makes a quick left turn you will feel as though
there is a sudden force pushing you to the right. This force is properly called a centrifugal force
because it is apparently directed away from the center of the curve. However, an observer looking
down upon the car from above will see that there is no outward force, only an inward force of the car
on your body making it accelerate inward. This is why a centrifugal force is called a fictitious force.
In physics, we seldom use the “f” word: centrifugal!
Conical Pendulums, Banking Airplanes, and Leaning Bicycles.
There are many situation where the centripetal force is provided by a horizontal component of a force
that also supports the object against gravity. For example, a ball on the end of a string can be swung in
a continuous circle called a conical pendulum. This is simply a pendulum that is swinging in two
planes at once. The tension force both holds the mass up against gravity, it also accelerates it toward
the center of the circle. A similar situation is provided with an airplane, with the lift force of the air on
the wings replacing the tension force. For a leaning bicycle or motorcycle it is the force pointing from
the tire's contact patch up through the center of gravity of the cycle that provides both the upward force
and the centripetal force. Usually, problems of this sort are interested in the angle the force deviates
from the vertical, sometimes called the bank angle.
Here's an example:
A airplane makes a turn of radius 400m at a speed of 30.0m/s. Find the bank
angle of the airplane, measured from vertical;
We draw a FBD of the situation with the airplane flying away from us:
Note that the vertical component of the lift vector is what holds the airplane
up against gravity, while the horizontal component of the lift vector provides
the centripetal force. Drawn as a vector diagram:
Fg
FL
Fh=Fc=mac
FL
Fv=Fg=mg
Note that the angle we are interested in is the angle between the vertical vector and the lift force vector.
The tangent of this angle is the ratio of the force of gravity to the centripetal force. Since both of these
involve the mass, the tangent is also the ratio of the centripetal acceleration to the acceleration due to
gravity!
Tanθ=opp/adj=Fc/Fg=mac/mg=ac/g
Of course, ac=v2/r, so
ac=302/400=2.25m/s
Tanθ=ac/g=2.25/9.8
θ=12.93 degrees.
Universal Gravitation:
Whenever we speak of rotational motion we usually also speak of orbital mechanics, the study of
astronomical orbits. Newton discovered that orbital motions follow exactly the same laws as motions
on earth. His big contribution was a greater understanding of gravitation.
Gravitation is governed by Newton's Law of Gravitation:
Fg=G(m1m2)/r2
Where Fg is the gravitational attractive force, G is the universal gravitational constant (6.67x10-11), m1
and m2are the two masses involved and r is the distance between the centers of the masses.
Gravitation is a vector quantity, so if you have multiple objects all pulling on one object you have to
add them up using vector methods.
In the case of planets, the gravitational force is the force causing the planet to move in a circular path,
so Fg is Fc. We also know that Newton's 2nd law and the definition of centripetal acceleration apply:
Fg=mac
ac=v2/r
Note that you can substitute the second equation into the first and then set it equal to the gravitation
equation. One of the m's can be divided out (this is the mass of the orbiting object) to get a single
equation that relates the mass of the object being orbited, the orbital radius, and the orbital velocity.
I would do this for you, but you'll learn more by doing it for yourself! Have fun!
Rotational Kinematics
We can also discuss the circular motion in terms of angular position, angular velocity, and angular
acceleration, just as we discussed linear position, velocity, and acceleration. The amazing thing is that
most of the same formulas apply to both linear and rotational situations. Of course, to make them look
right we have to use the correct variables. Here are the rotational variables and their corresponding
linear analogs.
Symbol
θ
ω
α
τ
I
L
Quantity
Linear
Analog
angular position
x
angular velocity
v
angular acceleration
a
torque
f
moment of inertia
m
angular momentum
p
Torque:
Torque is a twisting force applied to an object. When you twist a bottle cap you are applying a torque
to the cap. Note that you can twist the cap either way so torque is a vector quantity with a direction.
Conventionally, counterclockwise is positive and clockwise is negative. This convention comes from
the Right Hand Rule and the use of the Cartesian coordinate system.
The formula for torque is the following:
Pivot
τ=Flsinθ
l
θ
Where τ is torque, F is the applied force, l is
the lever arm, and θ is the angle between the
applied force and the lever arm
F
Moment of Inertia:
The moment of inertia is the rotational analog of mass or inertia. Like linear inertia it is dependent
upon mass, but it is also dependent upon the way the mass is arranged. The further the mass is from
the center of rotation, the greater the moment of inertia. The moment of inertia is a scalar, so if have
several moments of intertia from multiple objects you can just add 'em up!
The formula for moment of inertia of a small mass rotating about a point is:
I=mr2
Where I is the moment of inertia, m is the mass, and r is the distance from the pivot point to the mass.
This formula also works for thin rings, which are like hula-hoops rotating about their center point.
There are many other formulas for moments of inertia for various shapes, but this one will do for the
material we will be examining.
The great thing about rotational kinematics, as well as rotational momentum is that all of the formulas
we know and love convert readily to their rotational versions, simply by substituting the variables. For
example, the formula for linear position:
x=x0+v0t+(1/2)at2
converts to:
θ=θ0+ω0t+(1/2)αt2
As long as your angles are in radians, velocities in radians/second, and accelerations in radians/second
squared the formulas are plug-and-play for all kinematic situations! All the linear formulas convert in
exactly the same way!
Angular Momentum
Again, angular momentum acts just like linear momentum. The only big difference is the fact that
moment of inertia can be easily changed during the process whereas mass cannot. This change in
moment of intertia is apparent in sports and dance where an athlete or performer will alter their rate of
spin by rearranging their body's mass.
The formula for angular momentum is:
L=Iω
But, the formula for moment of inertia is
I=mr2
We can combine these into:
L=mr2ω
Now, notice that the moment of inertia can be markedly changed “on the fly” by changing the radius of
rotation of the mass. Since angular momentum is conserved, a small change in the radius will make a
large change in the angular velocity. A skater will start a slow rotation with her arms and legs
outstretched, giving herself a large moment of inertia. She will then pull her arms in and markedly
reduce her moment of inertia. Since her angular momentum must remain the same the result is that her
angular velocity (spin rate) increases dramatically. This is how skaters are able to spin so rapidly, and
just as important, how they can get out of the spin without hurting themselves!