Chapter 5 Notes
Circular Motion and Gravity
Centripetal Force
A curved path requires an inward pull.
centripetal force: the force needed to make an object follow a curved path
Centripetal force is the force perpendicular to the velocity of an
object moving along a curved path. The centripetal force is the force
directed toward the center of the curvature of the path.
uniform circular motion: the motion of an object that moves in a circle at
constant speed.
Why doesn't the definition say "constant velocity?"
the velocity of an object in circular motion continually changes in direction,
although its magnitude may remain constant.
Centripetal Acceleration
*It produces a change in direction without a change in speed.
For object’s experiencing circular motion, the acceleration that is the result
of an the object's change in velocity is called centripetal acceleration and
it points toward the center of the object's circular path.
ac = v2/r
I am not going to derive this equation!!
Period of Circular Motion
period: the time (T) needed by an object in uniform circular motion to
complete an orbit
Since the circumference is 2πr, then v = 2πr/T
Angular Measure
What units have we been measuring angles with?
What is the other system? radians
degrees
For rotational motion we will use radians.
How do you convert degrees to radians?
360 degrees = 2π radians
If a circle has a radius r and the arc cut by angle is s then the angle in
radians is θ = s/r
radians = arc length/radius
an angle of 1 radian has an arc length = radius
360° = 2π radians 1° = 0.01745 radians
57.3° = 1 radians
ex: A record with a diameter of 20.0 cm turns through an angle of 120°?
How far does a speck of dust on its rim travel?
θ = s/r
s = θr
s = 31.4 cm
what are the units for radians?
*a radian is equal to the radius. if you walk around a circle 1 radian you
have walked the same of the radius.
Circular Motion
An object can be traveling at a constant speed and still be accelerating.
Remember that acceleration is a vector quantity so if you change
direction you accelerate.
An object that moves in a circle at constant speed is said to have uniform
circular motion.
Revolution vs. Rotation
*When an object turns about an internal axis is the motion is called
rotation or spin (ex. an ice skater, wheels about an axis, a merry-go-round
*When an object turns about an external axis the motion is called
revolution. (ex: a person on a merry-go-around)
Earth
rotates on its axis every 24 hrs
revolves around the sun once every 365.24 days
Making a shift from linear to rotational motion.
-you already know the equations-you will just have to apply
them in different terms
Angular Speed
If a rotation body turns through the angle (distance)θ in the time t, its
average angular speed (ω) is ω = θ/t
this is analogous to v=d/t
units = rad/s
commonly you will here ω referred to in revolutions/sec or revolutions/min
rev/min→rad/s = (1rev/s)(2πrad/rev)(1min/60s)=2π/60 = 0.0105 rad / sec
*you can find the angular speed from the linear speed (or vice versa)
ex if you have an object moving with uniform linear speed v in a circle
radius r then:
so
recall
so
recall
so
the particle travels a distance (arc length) s in v∙t (v=d/t or v=s/t)
s = vt
θ = s/r
θ = vt/r
ω = θ/t
ω = (vt/r)/t
= vt/rt
=v/r
ω = v/r
Angular Acceleration
** the formulas for acceleration (rotational) are analogous to linear
acceleration
angular acceleration (α) is the change in angular speed over time
α = Δω/Δt
units are rad/s2
a particle moving in a circle (radius r) has an angular acceleration α and
also a tangent acceleration (at)
recall
Δω = Δv/r
and
α = Δω/Δt = α = Δv/(rΔt)
recall
at = Δv/Δt
so
α = at/r or at = αr
**tangential acceleration is NOT the same as centripetal acceleration (ac)
centripetal acceleration - produces a changes in direction w/o a change
in speed
- for an object in uniform circular motion ac
always points inward
so we have 3 accelerations to consider:
centripetal acceleration – produces a change in direction
angular acceleration tangential acceleration – produces a change in speed
ac and at are ALWAYS perpendicular
the formulas for linear motion are analogous to the kinetic formulas for
rotation
Linear
Angular
2
d = vot + ½ at
θ = ωot + ½αt2
v2 = vo2 + 2 ad
ω2 = ωo2 +2α θ
vavg = (vo + v)/2
ωavg = (ωo + ω)/2
vavg = d/t
ωavg = θ /t
a = (vo - v)/t
α = (ωo - ω)/t
Centripetal Force
*any force that causes an object to follow a circular path is called a
centripetal force
*the force perpendicular to the tangential velocity of an object moving
along a curve path. IT IS ALWAYS DIRECTED TOWARD THE CENTER
OF CURVATURE. (Centripetal actually means "center seeking")
Picture a ball being spun on a rope. the velocity is always tangent to the
curved path so the force is always toward the center. To get the ball to
curve you pull it toward center.
Centripetal force acts whenever rotational motion occurs.
Applications
1) Gravitation force provides the centripetal force that keeps bodies in
orbit
2) When a car goes around a bend the sideways friction between the
tires and the road provide centripetal force. If friction isn't great enough
the car slides sideways
Centripetal Force can be calculated using Newton's second law: F=ma
Fc =mac
ac = centripetal acceleration
* ac produces a change in direction without a change
in speed
*For an object in uniform circular motion ac points inward
recall:
accel = Δv/ Δt
and
v = d/t
so if we want a in terms of v and d …
(1) rearrange v = d/t → t=d/v
(2) substitute a = (vt-o)/(d/v)
(3) simplify a = v2/d
Here we use distance = radius so ac = v2/r
(4) so Fc = mac = mv2/r
example: Find the centripetal force needed by a 1200 kg car to make a
turn of radius 40m at 25 km/hr
m=1200 kg
r = 40 m v = 25km/hr =7 m/s
Fc = mv2/r
= (1200kg)(7m/s)2
40m
Fc =1470 N
Assuming the road is level, find the minimum coefficient of static friction
between the car's tires and the road that will permit the turn to be made.
Frictional force is given by:
Ff = µFN FN = the car's weight = mg = 1200kg(9.8m/s2) = 11760N
Ff = 1470N (Fc )
µ = Ff/FN = 1470N / 11760N = 0.125
The coefficient of friction does not depend on the car's mass since,
µ = Ff = Fc = mv2/r = v2/r = v2
FN FN
mg
g
gr
because the required value of µ depends on v2, high speed turns on a
level road can be dangerous. In this example increasing the speed to 60
km/h (or 37 mi/hr) increases the needed µ to 0.72, which is too much for a
wet road so the car will skid
DO NOT CONFUSE CENTRIPETAL FORCE WITH
CENTRIFUGAL FORCE
Imagine riding in a car when it suddenly makes a left turn, you may feel
an outward force toward the right of the car. That b/c a passenger would
continue in a straight line if the seatbelt or door were not there (applying
an inward centripetal force). But a passenger feels an outward force.
THIS OUTWARD FORCES IS CALLED CENTRIFUGAL FORCE.
Centrifugal means "center fleeing"
examples of "feeling" centrifugal forces:
(1) When a washing machine goes into the spin cycle, what happens?
The tub rotates at high speed. The tub exerts a centripetal force inward
on the clothes, however, it seems like the the water is pushed out. But
nothing actually pushes out – The tub pushes in on the clothes, but the
holes in the tub prevent the tub from exerting the same force on the water.
(2) The "Rotor" spinning amusement ride
as it spins your velocity is tangent to the curve – if not for the walls of the
ride – you would continue in that tangent path. Therefore it seems like
you are being pushed out – But the only true force is inward.
(3) Swing a bucket of water over your head
Banked roads and tracks
Usually friction is enough to keep a car on the road. However, if the car's
speed is high (or the road is slippery) the friction force may not be
enough. Highway curves and especially race tracks are banked so that
the road tilts inward. This helps to keep the car pressing on the road, and
centripetal force great enough to keep the cars from skidding off the road.
*the same idea applies to airplanes banking to make turns