Gravity

Kinematics §

• In free fall, objects accelerate due to gravity. We denote the acceleration due to gravity as .

  • If a freely falling object passes a given point at two different times, once moving upward and once moving downward, its speed will be the same at both times.
  • When the acceleration of an object is constant, its position as a function of time is given by a quadratic equation.

• The motion of a projectile is a combination of motion with constant velocity in the horizontal -direction and motion with constant downward acceleration in the vertical -direction.

• Velocity and Acceleration are perpendicular only at the peak of the trajectory.

Dynamics §

• The Law of Gravitation states that every object attracts every other objects with a gravitational force that follows

  Where is the gravitational constant and is the distance between the two objects with masses and .

  • If the objects are not point masses, we can approximate by using the center of mass.

• Gravity can be formulated as acting on a field. One object emits this field at all points in space, and the force acts on an object at a particular point.

Orbits §

• An object orbits around a body if it is falling towards the body constantly in such a way that it maintains a certain distance away from the body.

  The speed an object needs to maintain an orbit of radius around an object with mass is calculated as

  This derives from the Law of Gravitation and Newton’s Second Law. In particular, we take the mass of the object and the centripetal acceleration . Both forces must balance since the object stays in orbit (i.e., does not decrease in height)

• The Period of a circular orbit is related to the speed in that the speed is the distance traveled in one revolution.

  • The period can also be derived by using the calculation of orbit speed above

    We can generalize the above for elliptic orbits. Let be the semi-major axis

• Kepler’s Laws of Planetary Motion describes the orbits of planets around the sun.

  • Law 1: The orbit of a planet is an ellipse with the Sun as one of the two foci.
  • Law 2: A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. The sector velocity is the velocity of this sweeping line. It is calculated as
    Kepler’s Second Law states that sector velocity is constant.
  • Law 3: The square of the period of the orbit is proportional to the cube of the length of the semi-major axis of the orbit.

• The laws of planetary motion are a consequence of the Laws of Universal Gravitation

  • For an object acted on by an attractive force proportional to , the only possible closed orbits are a circle or an ellipse. A generalization of this is that all orbits are conic sections (open trajectories are hyperbolic or parabolic).
  • The second law follows from the conservation of angular momentum.
  • The third law follows from the calculation above on the orbit’s period.

Links §

• University Physics with Modern Physics 15th Edition By Freedman - Ch. 13