Transcript Circular Motion - strikerphysics11

Circular Motion
Chapter 7 (already)
Polar Coordinates
We commonly use
Cartesian or
rectangular coordinate
system where (x, y)
identifies a point in two
dimensions.
We could equivalently
use polar coordinates
to identify a point (r, θ)
Converting Between Coordinate
Systems
Given r and θ,
Given x and y,
X = r·cosθ
r = (x2 + y2) ½
Y = r·sinθ
θ = tan -1 (y/x)
This converts from a
polar coordinate to a
rectangular
coordinate
This converts from a
rectangular
coordinate to a polar
coordinate
Circular Motion
Imagine a point
moving along the
circle at constant
speed.
Which variables
change and which
stay the same?
Circular Motion
The angle, θ, changes
but the radius does not
change.
The coordinates x- and yboth change.
The speed does not
change but the velocity
does change (why?)
Angle Measure
Angles may be
measured in degrees
or in radians.
360° = 2π
The length of an arc
subtending a circle of
radius r is given by
S = r·θ
Examples
A spectator is standing at the center of a circular
running track and observe a runner start practice
256 m due east of her position. The runner
runs on the track in a counter clockwise direction
to the finish line, located due north of the
observer. What was the distance of the run?
A sailor sights a distant ship and finds that it
subtends an angle of 1.15°. He knows from the
shipping charts that the tanker is 150 m in
length. How far away is the tanker?
Angular Speed / Velocity
Angular speed, ω, is defined as the rate of
change of θ.
ω = Δθ/ Δt [radians/sec]
and θ = ωt
Then
s = rθ becomes s = r (ωt)
And
v=rω
Speeds in Circular Motion
Angular velocity is denoted with ω, omega,
and the direction is at a right angle to the
plane of the circular motion (use right hand
rule)
Tangential velocity is denoted with v,
velocity.
Radians~!
Note that ω is measured in radians/sec.
The formulas involving ω are true only if θ
(angular displacement) is measured in
radians since only then is s = rθ a true
statement.
Period and Frequency
The Period, T, is the time necessary for one
complete cycle. Period, T, is measured in
[seconds].
The frequency, f, is the number of cycles that
happen in one second. Frequency is measured
in [cycles/second] or [Hertz]
T = 1/f
Angular speed, ω = 2π/T = 2πf
Examples
A merry go round at its constant operational
speed makes one complete rotation in 45
seconds. Two children are on horses, on at 3.0
m from the center and one 6.0 m from the
center. Find the angular speed and the
tangential speed for each child.
A CD rotates in the player at a constant speed of
200 rpm. What are the CD’s a) frequency and b)
Period of revolution?
Centripetal Acceleration
An object in Uniform Circular Motion
(constant speed) experiences a
Centripetal Force ( and therefore a
centripetal acceleration) that is pointed
towards the center.
ac = v2/r = (rω)2/r = rω2
Fc = m(v2/r) = mrω2
Examples
A lab centrifuge operates at a rotational
speed of 12,000 rpm. A) What is the
magnitude of the centripetal acceleration
of a red blood cell at a radial distance of
8.0 cm from the centrifuge’s axis of
rotation? Compare this with g.
A ball attached to a string is swung with
uniform motion in a horizontal circle over a
person’s head. If the string breaks, what
will be the trajectory of the ball?
More Examples…
A car approaches a level circular curve with
radius of 45.0 m. If the concrete pavement is
dry, what is the maximum (constant) speed at
which the car can negotiate the curve?
Two masses, m1 = 2.5 kg and m2 = 3.5 kg are
connected by light strings and are in uniform
circular motion on a horizontal frictionless
surface, with r1 = 1.0m and r2 = 1.3m. The
forces acting an the masses are T1 and T2
respectively. Find the centripetal acceleration
and the tangential speed of mass 2 and mass 1.
Angular Acceleration
When circular motion isn’t uniform, angular
velocity changes.
α , alpha, is used to denote angular acceleration.
We consider only the case of constant angular
acceleration so
αav = α = Δω/t
Then Δω = αt and ωf = ωi + αt
Tangential Acceleration
If the rate of angular velocity changes,
then the object undergoes angular
acceleration and tangential acceleration:
at = Δv/t = Δ(rω)/t = rα
at = rα
Linear and Rotational
Kinematics
ωavt
X = vav t
θ=
Vav = (vf + vi)/2
ωav = (ωf+ωi)/2
Vf = vi + at
ωf = ωi + αt
Xf = xi + vit + ½ at2
θf =θi + ωit + ½ αt2
Vf2 = vi2 + 2aΔx
ωf2 = ωi2 + 2αΔθ
Examples
A CD accelerates uniformly from rest to its
operational speed of 500 rpm in 3.0 sec. What is
the angular acceleration of the CD during this
time? If the CD comes to a stop in 4.0 sec,
what is the angular acceleration during that part
of the motion?
A microwave oven has a 30 cm rotating plate.
The plate accelerates from rest to a uniform rate
of 0.87 rad/s2 for 0.50 sec before reaching its
constant operational speed. How many
revolutions does the plate make before reaching
its operational speed? What are the operational
angular speed and tangential speed at its rim?
Newton’s Law of Gravitation
Newton wondered whether the force
pulling apples to the ground was the same
force pulling the moon towards the Earth.
He recognized that, although the moon
orbits the Earth in a circular path, that the
moon really is ‘falling’ towards the Earth
Newton’s Cannon Thought
Experiment
Gravitational Force
Gravitational Force
Is always attractive
Is proportional to the product of the
masses
Is inversely proportional to the square of
the distance between the masses
G = 6.67 X 10 -11 Nm2/kg2
Examples
Compute the gravitational force between
Earth and the Moon. Mm = 7.4 x 1022 kg,
ME = 6.0 X 1024 kg, rEM = 3.8 X 108 m.
Find the acceleration due to gravity acting
on an object with mass Mo near the Earth’s
surface. rE = 6.4 X 106m
Satellite Example
Some communication and weather
satellites are launched into circular orbits
above the Earth’s equator so they are
synchronous with Earth’s rotation. At
what altitude are these satellites?
RE = 6.4 X 106 km;
ME = 6.0 X 1024 kg,
T = 365 days