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Uniform circular motion is motion in a circular path at constant speed.
Under uniform circular motion, angular and linear quantities have simple relationships. When objects rotate around some axis, each point of the object follows a circular arc. The angle of rotation is the amount of rotation and is analogous to linear distance. We define the angle of rotation Δθ to be the ratio of the arc length to the radius of curvature:
Δθ = Δs / r
Angle θθ and arc length ss: The radius of a circle is rotated through an angle Δθ . The arc length Δs is described on the circumference.
We define the angular velocity ωω as the rate of change of an angle. In symbols, this is ω=Δθ / Δt, where Δθ produces an angular rotation in a time Δt . From the relationship of s and ( Δs=rΔθ ), we see:
v= Δs / Δt =r Δθ / Δt = rω
Under uniform circular motion, the angular velocity is constant. The acceleration can be written as:
ac = dv / dt = ω dr / dt = ωv = rω^2 = v^2 / r
This acceleration, responsible for uniform circular motion, is called centripetal acceleration.
Any force or combination of forces can cause centripetal or radial acceleration. Just a few examples are the tension in a string on a tether ball, the force of Earth's gravity on the Moon, the friction between skates and the floor of a rink, the force of an inclined roadway on a car, and the forces in the tube of a spinning centrifuge.
Any net force that causes uniform circular motion is called a centripetal force. The direction of a centripetal force is toward the center of curvature, the same as the direction of centripetal acceleration. According to Newton's second law of motion, the net force is the acceleration times mass. For uniform circular motion, the acceleration is the centripetal acceleration: a=ac>. Thus, the magnitude of the centripetal force Fc is:
Fc = mac = m v^2 / r = mrω^2
Newton's universal law of gravitation states that every particle in the universe attracts all other particles with a force along a line joining them. The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. For two bodies having masses mm and MM with a distance rr between their centers of mass, the equation for Newton's universal law of gravitation is:
F = G(mM / r^2)
The gravitational force is responsible for artificial satellites orbiting the Earth. The Moon's orbit around the Earth, and the orbits of planets, asteroids, meteors, and comets around the Sun are other examples of gravitational orbits. Historically, Kepler discovered his three laws long before Newton's time. Kepler devised his laws after a careful study of a large number of meticulously recorded observations of planetary motion made by Tycho Brahe.
The orbit of each planet around the Sun is an ellipse with the Sun at one focus.
Each planet moves so that an imaginary line drawn from the Sun to the planet sweeps out equal areas in equal times.
The ratio of the squares of the periods of any two planets around the Sun is equal to the ratio of the cubes of their average distances from the Sun.
Kepler's Second Law: The shaded regions have equal areas. It takes equal times for mm to travel from A to B , from C to D , and from E to F . Mass mm moves faster when it is closer to M . Kepler's second law was originally devised for planets orbiting the Sun, but it has broader validity.
Ellipses and Kepler's First Law: An ellipse is a closed curve such that the sum of the distances from a point on the curve to the two foci ( f1 and f2 ) is a constant. You can draw an ellipse as shown by putting a pin at each focus, then putting a string around a pencil and the pins and drawing a line on paper. A circle is a special case of an ellipse in which the two foci coincide (so any point on the circle is the same distance from the center). b) For any closed gravitational orbit, m follows an elliptical path with M as one focus. Kepler's first law establishes this fact for planets orbiting the Sun.
Kepler's third law is equivalent to:
T is the period (time for one orbit) and r is the average radius. We will derive Kepler's third law, starting with Newton's laws of motion and his universal law of gravitation. We will assume a circular (not elliptical) path for simplicity.
Consider a circular orbit of a small mass mm around a large mass M , satisfying the two conditions outlined at the beginning of this section. Gravity supplies the centripetal force to mass m . Therefore, for uniform circular motion:
G (mM/r^2) = mac = m(v^2/r)
The mass m cancels out, yielding:
G (mM/r^2) = v^2
Now, to arrive at Kepler's third law, we must plug point T into the equation. By definition, period T is the time for one complete orbit. Now the average velocity v is the circumference divided by the period:
v = 2πr/T
Substituting this into the equation above gives:
G(M/r) = 4π^2r^2 / T^2
Solving for T^2 yields:
T^2 = (4π^2 / GM)r^3
Since T^2 is proportional to r^3 , their ratio is constant. This is Kepler's third law.
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We recommend visiting the following materials for further knowledge or understanding on the topic:
1. Uniform Circular Motion and Gravitation6. The angle of rotation is the amount of rotation and is analogous to linear distance.
7. The gravitational force between the Earth and the Moon acts as a centripetal force that maintains its orbit.
8. Any force or combination of forces can cause centripetal acceleration.
9. F = G(mM / r^2)
10. Any net force that causes uniform circular motion is called a centripetal force.
References:
1. Libretexts. (2022, 31 octubre). 5.1: Introducción a la UCM y Gravitación. LibreTexts Español. https://espanol.libretexts.org/Bookshelves/Fisica/Libro%3A_Fisica_(sin_limites)/5%3A_Movimiento_Circular_Uniforme_y_Gravitaci%C3%B3n/5.1%3A_Introducci%C3%B3n_a_la_UCM_y_Gravitaci%C3%B3n https://espanol.libretexts.org/Bookshelves/Fisica/Libro%3A_Fisica_(sin_limites)/5%3A_Movimiento_Circular_Uniforme_y_Gravitaci%C3%B3n/5.1%3A_Introducci%C3%B3n_a_la_UCM_y_Gravitaci%C3%B3n
2. Libretexts. (2024, 1 octubre). 6: Uniform Circular Motion and Gravitation. Physics LibreTexts. https://phys.libretexts.org/Bookshelves/College_Physics/College_Physics_1e_(OpenStax)/06%3A_Uniform_Circular_Motion_and_Gravitation https://phys.libretexts.org/Bookshelves/College_Physics/College_Physics_1e_(OpenStax)/06%3A_Uniform_Circular_Motion_and_Gravitation
3. Univeristy of Nebraska. (s. f.). Physics, Chapter 6: Circular Motion and Gravitation. https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1141&context=physicskatz https://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1141&context=physicskatz
4. Monash Umiversity. (s. f.). Gravitational Circular Motion. https://www.monash.edu/student-academic-success/physics/forces/gravitational-circular-motion https://www.monash.edu/student-academic-success/physics/forces/gravitational-circular-motion
5. Professor Dave Explains. (2017a, marzo 17). Uniform Circular Motion and Centripetal Force [Vídeo]. YouTube. https://www.youtube.com/watch?v=SZj6DuB0vvo https://www.youtube.com/watch?v=SZj6DuB0vvo
6. The Organic Chemistry Tutor. (2023, 11 octubre). Uniform Circular Motion Formulas and Equations - College Physics [Vídeo]. YouTube. https://www.youtube.com/watch?v=IawY86XveQE https://www.youtube.com/watch?v=IawY86XveQE
7. CrashCourse. (2016, 12 mayo). Uniform Circular Motion: Crash Course Physics #7 [Vídeo]. YouTube. https://www.youtube.com/watch?v=bpFK2VCRHUs https://www.youtube.com/watch?v=bpFK2VCRHUs