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Unformatted text preview: Circular Motion; Gravitation
Chapter Outline GENERAL PHYSICS
Kinematics of Uniform Circular Motion PHY 302K Dynamics of Uniform Circular Motion
Chapter 5: Circular Motion; Gravitation
Highway Curves, Banked and Unbanked
Nonuniform Circular Motion Maxim Tsoi Newton’s Law of Universal Gravitation
Kepler’s Laws Physics Department,
The University of Texas at Austin
http://www.ph.utexas.edu/~tsoi/
302K  Ch.5 302K  Ch.5 Uniform Circular Motion Uniform Circular Motion A car moving in a circular path with constant speed Period • Even though the car moves at a constant
speed, it still has an acceleration! • Motion of a particle moving with constant
speed in a circle of radius r is described in direction of the car’s velocity vector changes • Centripetal acceleration: aC terms of the period T 2 v
r time required for one complete revolution v f vi v
Average acceleration a t f ti
t Similar triangles v
v T r 2 r
v r v v r
a t r t 302K  Ch.5 302K  Ch.5 Uniform Circular Motion Nonuniform Circular Motion Newton’s 2nd law applied to uniform circular motion Radial and tangential forces • Ball of mass m is tied to a string of • particle moves with varying speed in a length r and is being whirled at constant circular path particle’s acceleration has speed v in a horizontal circular path on a radial and tangential components frictionless table a ar at • The ball experiences centripetal
acceleration ac the force acting on the particle also v2
r has two components F F F • The string exerts on the ball a radial
force Fr r v2
Fr ma c m
r t • Radial force centripetal acceleration
• Tangential component tangential • If string breaks the ball moves along a acceleration (change of speed) straightline path tangent to the circle
302K  Ch.5 302K  Ch.5 1 Tangential and Radial Acceleration Newton’s Law of Universal Gravitation Radial and tangential components of acceleration • A particle moves along a smooth curved Unification of “earthly” and “heavenly” motions Every particle in the Universe attracts every other particle with a
force that is directly proportional to the product of their masses and
inversely proportional to the square of the distance between them path where the velocity changes both in
direction and in magnitude
• The velocity vector is always tangent to Fg G the path
• The acceleration vector can be resolved
into two components
2 v
ar r v at t a a t ar m1m 2
r2 • G is the universal gravitational constant = 6.6731011 Nm2/kg2 Radial component ar arises mm
ˆ
F12 G 1 2 2 r12
r from the change in direction of the
velocity vector Tangential component at F21 F12 causes the change in speed of the
particle 302K  Ch.5 302K  Ch.5 Newton’s Law of Universal Gravitation Newton’s Law of Universal Gravitation Unification of “earthly” and “heavenly” motions Unification of “earthly” and “heavenly” motions Every particle in the Universe attracts every other particle with a
force that is directly proportional to the product of their masses and
inversely proportional to the square of the distance between them Every particle in the Universe attracts every other particle with a
force that is directly proportional to the product of their masses and
inversely proportional to the square of the distance between them Fg G m1m 2
r2 • Gravitational force is a field force always exist between two particles
regardless of the medium that separates them Fg G
Fg G MEm ma
r2 a G 2 • Fg~1/r2 decreases rapidly with increasing separation
• a finitesize spherically symmetric mass distribution produces the same
gravitational force as if the entire mass were concentrated at its center MEm
2
RE 302K  Ch.5 ME
r2 2 2
a M 1 rM RE 6.37 106 m 2.75 104 2
g 1 RE rM 3.84 108 m a M 2.75 10 4 9.8m / s 2 2.70 103 m / s 2 v 2 2rM / T 4 2 3.84 108 m
aM 2.72 10 3 m / s 2
2
rM
rM
2.36 106 s
2 Fg G m1m 2
r2 302K  Ch.5 Measuring the Gravitational Constant FreeFall Acceleration Experiment by Henry Cavendish in 1798
• Two spheres (each of mass m) fixed to the ends of a light horizontal
rod suspended by a thin metal wire g and the Gravitational Force • The magnitude of the force acting on a freely falling object of
mass m near the Earth’s surface: • Two large spheres (each of mass M) are placed near the small ones mg G MEm
2
RE g G ME
2
RE • For an object located a distance h above the Earth’s surface: Fg G
• The attractive force between smaller and larger spheres causes the MEm
MEm
G
2
r2
RE h g G R ME E h 2 rod to rotate and twist the wire • g decreases with increasing altitude! • The angle of rotation is measured by deflection of light beam for
different masses at various separations
302K  Ch.5 302K  Ch.5 2 Kepler’s Laws and the Motion of Planets Kepler’s Laws and the Motion of Planets Kepler’s analysis of planetary motion is summarized in three statements First Law All planets move in elliptical orbits with the Sun at one focus
(1)
All planets move in elliptical orbits with the Sun at one focus ec a • Eccentricity of an ellipse Earth: 0.017; Pluto: 0.25; Comet Halley: 0.97 (2)
The radius vector drawn from the Sun to a planet sweeps out
equal areas in equal time intervals
(3)
The square of the orbital period of any planet is proportional to
the cube of the semimajor axis of the elliptical orbit • Aphelion the point where the planet is farthest away
from the Sun (apogee for an object orbiting the Earth)
• Perihelion the point nearest the Sun (perigee for an
object orbiting the Earth) 302K  Ch.5 302K  Ch.5 Kepler’s Laws and the Motion of Planets Kepler’s Laws and the Motion of Planets Second Law Third Law The radius vector drawn from the Sun to a planet sweeps out equal
areas in equal time intervals The square of the orbital period of any planet is proportional to the
cube of the semimajor axis of the elliptical orbit GM S M P M P v 2 r2
r • Newton’s 2nd law:
• consequence of angular A const
t • The period: 4 2
T2 GM
S 302K  Ch.5 v • The orbital speed: momentum conservation 3
a K S a 3 2r
T 4 2
T2 GM
S KS 3
r K S r 3 4 2 2.97 1019 s 2 / m 3
GM S 302K  Ch.5 SUMMARY
Circular Motion; Gravitation
• Centripetal acceleration: ac v 2 / r • Newton’s law of universal gravitation:
• Kepler’s laws of planetary motion: Fg G m1 m 2
r2 (1) All planets move in elliptical orbits with the Sun at one focus
(2) The radius vector drawn from the Sun to a planet sweeps out
equal areas in equal time intervals
(3) The square of the orbital period of any planet is proportional
to the cube of the semimajor axis of the elliptical orbit 302K  Ch.5 3 ...
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 Spring '09
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 Physics, Circular Motion

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