AP Physics - Uniform Circular Motion

• Rotation

  • Spinning based on internal axis

• Circular Motion

  • Spinning based on external point

Centripetal Force

• Center seeking

  • Not centrifugal

    • Inertia of the object

• ΣF_c

• Can be any type of force

  • Moving in a circle

History

• 384 B.C.E. - 322 B.C.E. Aristotle and Plato

  • Geocentric

    • Earth Centered

  • Believed the heavens were small remote objects in motion around the earth that moved with constant angular rate

310 B.C.E - 230 B.C.E (Aristarches)

• Believed that the sun was the center of the universe

  • Heliocentric

    • Sun centered

  • Believed there were spheres for the stars, planets, and Earth

    • celestial sphere was motionless

    • earth rotates once on an axis of its own

    • planets moved in circular paths around the sun

150 C.E. Claudius Ptolemy

• Believed both systems could be used in describing motion

  • Preferred geocentric theory because it fit the causes of motion of the planets, stars, and sun

  • Believed the earth was the center of the universe but not the center of all the heavenly circles

Ptolemy Model

• Developed a very clever and rather accurate procedure for predicting the positions of each planet on a geocentric model

  • Constructed a model out of circles and three other geometrical devices that would each provide for variations in the rate of angular motion as seen from Earth

    • Eccentric

    • Epicycle

    • Equant

• Successes:

  • Found a combination of motions that gave more accurate predictions on positions of planets to withing 2° over a long period of time

• Limitation:

  • Not possible to calculate the period or the size of each planet’s orbit

1473 - 1543 (Nicolaus Copernicus)

• Adopted the heliocentric system with the following assumptions

  • No precise geometrical circles

  • Center of the earth was not the center of the universe, but the cause of gravitation for the moon

  • The distance from the earth to the sun is very small in comparison to the distance to the stars

  • The earth has more than one motion

    • It rotates on an axis

    • It revolves around the sun

• Successes (Copernicus was able to calculate):

  • The period of motion of each planet

  • The size of each planet’s orbit in comparison to the earth’s orbit

• Limitations:

  • Founded on inaccurate data

1546 - 1601 (Tycho Brahe)

• Believed in the geocentric system

• Recorded positions of the planets in trying to prove the geocentric theory

• Collected astronomical data and instruments that would result in more accurate positions of planets and stars

1571 - 1630 (Johannes Kepler)

• Assistant to Brahe and believed in a heliocentric system

• Used sun centered system to explain all Brahe’s data

  • Plotted the orbits of the planets and saw that these orbits looked like flattened circles

1666 (Isaac Newton)

• Falling apple made him think of the moon

  • Moves in circular path

  • Moon is accelerating because changing direction

  • Acceleration is caused by a force

  • Hypothesized that force was same force of gravity that caused an apple to fall

• Moon fall should be in direct proportion to fall of apple on earth

• Should relate to distances from earths center

  • Moons distance was 60 times greater than apple on earth. Gravity should be diluted by distance

  • Apple falls 4.9m in first second on earth

  • Moon falls 1.4mm in first second of orbit

    • 4.9/(60)²

• F_g ∝ 1/d²

• Gravitational force depends on both mass of object and mass of planet

  • F_g ∝ m_{1 *} m₂

• Hypothesized that force was proportional to the mass

• Used mathematics to show that the force for elliptical paths must be inversely related to the distance squared

• Showed that Force was direction along a line connecting the centers of the two bodies

• So confident with his work he wrote the Universal Law of Gravity Equation

  • F_g = G * (m₁ m₂) / d²

• He proved mathematically that F_g was the cause of elliptical paths of the planets by deriving Kepler’s 3rd Law from the universal gravity equation

1731 - 1810 (Henry Cavendish)

• Developed a torsion balance sensitive enough to measure gravitational attraction between two masses on Earth

  • Gravitational constant

    • G = 6.67 × 10^-11 Nm²/Kg²

  • Gravitational force is an inverse square law

Kepler’s Laws

Kepler’s 1st Law

• Law of Ellipses

  • The paths of the planets are ellipses with the sun at one foci

Kepler’s 2nd Law

• Law of Areas

  • An imaginary line from the sun to a planet sweeps out equal areas in equal time intervals.

    • Planets move fastest when closest to the sun, slowest when farthest away

Kepler’s 3rd Law

• Law of Periods

  • The ratio of the squares of the periods of any two planets revolving about the sun is equal the ratio of the cubes of their average distances from the sun