Pre-AP Circular Motion Motion 1 Circular Motion Rotational Motion 2 Uniform Circular Motion An...
3
Rotational Motion 1 Circular Motion Rotational Motion 2 Uniform Circular Motion An object that moves in a circle at constant speed is said to experience uniform circular motion. • The magnitude of the velocity remains constant. • The direction of the velocity is continuously changing as the object moves around the circle. • The object is accelerating because there is a change in velocity. This acceleration is called centripetal acceleration and it points towards the center of the circle. Rotational Motion 3 Centripetal Acceleration v 1 v 2 !v F E D v 1 v 2 A B C r a rad = "v "t "DEF # "ABC "v = v 2 # v 1 "v v = AB r so "v v or v 2 = "v + v 1 and AB = d = v "#t = a c = v 2 r = v 2 " #t r " #t and "v "t = v " #t r Rotational Motion 4 Centripetal Acceleration a c = v t 2 r • This component always points towards the axis of rotation. v a c v v t • The centripetal acceleration is always perpendicular to tangential motion. Rotational Motion 5 Universal Law of Gravitation Every particle in the universe attracts every other particle with a force proportional to the product of their masses and inversely proportional to square of the distance between them. Rotational Motion 6 Newton’s Law of Gravitation Newton proposed that any two masses were attracted by a gravitational force (inverse square law). Where: F = gravitational force (N) F g = G m 1 m 2 r 2 m = mass of body (kg) r = distance between the center of m 1 and m 2 (m) G = 6.67 x 10 -11 N ! m 2 kg 2 " # $ % & '
Transcript of Pre-AP Circular Motion Motion 1 Circular Motion Rotational Motion 2 Uniform Circular Motion An...
Rotational Motion 1
Circular Motion
Rotational Motion 2
Uniform Circular Motion
An object that moves in a circle at constant speed issaid to experience uniform circular motion.
• The magnitude of the velocity remains constant.• The direction of the velocity is continuously
changing as the object moves around the circle.• The object is accelerating because there is a
change in velocity.
This acceleration is called centripetal accelerationand it points towards the center of the circle.
Rotational Motion 3
Centripetal Acceleration
v1v2
!v
F
E
D
v1
v2
A
B
C
r
!
arad ="v"t
!
"DEF # "ABC
!
"v = v2 # v1
!
"vv
=ABr
!
so "vv
!
or v2 = "v + v1
!
and AB = d
!
= v " #t
!
= ac
!
=v 2
r
!
=v 2 " #tr " #t
!
and "v"t
!
=v " #tr
Rotational Motion 4
Centripetal Acceleration
!
ac =vt2
r
• This component always points towards the axis of rotation.
!
v a c
!
v v t
• The centripetal acceleration is always perpendicularto tangential motion.
Rotational Motion 5
Universal Law of Gravitation
Every particle in the universe attracts every otherparticle with a force proportional to the product oftheir masses and inversely proportional to squareof the distance between them.
Rotational Motion 6
Newton’s Law of GravitationNewton proposed that any two masses were attractedby a gravitational force (inverse square law).
Where:F = gravitational force (N)
!
Fg = Gm1m2r 2
m = mass of body (kg)r = distance between the center of m1 and m2 (m)
G = 6.67 x 10-11 N !m2
kg2"
# $
%
& '
Rotational Motion 7
Newton’s Law of Gravitation
!
Fg1 ,2 = Fg2 ,1 = Gm1m2r 2
r
!
m1
!
m2
!
v F g2 ,1
!
v F g1 ,2
Rotational Motion 8
Weight
The acceleration of objects due to Earth’s gravitationcan be found by using the inverse square law andNewton’s second law.
!
Fg = G MmR2
So the acceleration due to gravity at the surface ofa planet is:
!
g =GMR2
!
= mg
where:M = mass of the planet (kg)R = radius of the planet (m)
Rotational Motion 9
Weight
At a point above the earth’s surface a distance r fromthe center of the earth the weight of a body is:
!
Fg = G Mmr 2
M
R
!
r = R + d
dm
Rotational Motion 10
Motion of Satellites
A satellite in an orbit that is always the same heightabove a planet moves with uniform circular motion.Using Newton’s second law:
!
G Mmr 2
=mv 2
rThe orbital speed is therefore:
!
v =GMr
!
Fg = mac
Rotational Motion 11
Motion of Satellites
!
v =2"rT
Therefore the period is:
!
T =2"rv
!
= 2" r 3
GM
!
= 2"r rGM
For circular orbits, the period T of the satellite isrelated to speed v.
Rotational Motion 12
Kepler’s Laws of Planetary Motion1.) The paths of the planets are ellipses with the center of
the Sun at one focus.2.) An imaginary line from the Sun to a planet sweeps out
equal areas in equal time intervals. Thus, planets movefastest when closest the Sun, slowest when farthestaway.
3.) The ratio of the squares of the periods of any twoplanets revolving about the Sun is equal to the cubes oftheir average distances from the Sun.
!
TaTb
"
# $
%
& '
2
=rarb
"
# $
%
& '
3
Rotational Motion 13
Kepler’s Third Law (Newton’s version)
Squaring both sides:
This equation is Kepler’s third law.
!
T 2 =4" 2
GM#
$ %
&
' ( r 3
Recall for circular orbits:
!
T = 2" r 3
GM
Rotational Motion 14
Kepler’s Third Law (Newton’s version)
!
T 2 =4" 2
GM#
$ %
&
' ( r 3
!
Ta2 =
4" 2
GM#
$ %
&
' ( ra
3
!
and Tb2 =
4" 2
GM#
$ %
&
' ( rb
3
!
Ta2
Tb2 =
4" 2
GM#
$ %
&
' ( ra
3
4" 2
GM#
$ %
&
' ( rb
3
!
so TaTb
"
# $
%
& '
2
=rarb
"
# $
%
& '
3
15
Vertical Motion of a Mass on a Cord
Fg
Fg
FgFg
Top
Bottom
m
m
m
mac
ac
ac
ac
!
v F net = mv a
!
T " Fg =mac
!
T =mac + Fg
!
T =m ac + g( )
T
T
!
v F net = mv a
!
T + Fg =mac
!
T =mac " Fg
!
T =m ac " g( )
Rotational Motion 16
Roller Coaster Loops
!
v F net = mv a
!
FN " Fg =mac
!
FN = Fg +mac
!
FN =mg +mac
!
FN =m g + ac( )!
FN
!
Fg!
ac
!
FN
!
Fg
!
ac
!
v F net = mv a
!
Fg + FN =mac
!
FN =mac " Fg
!
FN =mac "mg
!
FN =m ac " g( )
Rotational Motion 17
Ferris Wheel
!
v F net = mv a
!
Fg " FN =mac
!
FN = Fg "mac
!
FN =mg "mac
!
FN =m g " ac( )!
FN
!
Fg
!
ac
!
FN
!
Fg!
ac
!
v F net = mv a
!
FN " Fg =mac
!
FN = Fg +mac
!
FN =mg +mac
!
FN =m g + ac( ) Rotational Motion 18
Car on a Curved Road
!
ac
!
Ff
!
Ff
!
Ff!
FN
!
Fg!
m
(View from above) (Side view)
!
v F net = mv a
!
Ff =mac
!
µFN =mac
!
µmg =mac
!
µg = ac
!
F" y =ma
!
FN " Fg = 0
!
FN = Fg =mg
!
F" x =ma
