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13.S: Gravitation (Summary)

Key Terms

 action-at-a-distance force type of force exerted without physical contact aphelion farthest point from the Sun of an orbiting body; the corresponding term for the Moon’s farthest point from Earth is the apogee apparent weight reading of the weight of an object on a scale that does not account for acceleration black hole mass that becomes so dense, that it collapses in on itself, creating a singularity at the center surround by an event horizon escape velocity initial velocity an object needs to escape the gravitational pull of another; it is more accurately defined as the velocity of an object with zero total mechanical energy event horizon location of the Schwarzschild radius and is the location near a black hole from within which no object, even light, can escape gravitational field vector field that surrounds the mass creating the field; the field is represented by field lines, in which the direction of the field is tangent to the lines, and the magnitude (or field strength) is inversely proportional to the spacing of the lines; other masses respond to this field gravitationally bound two object are gravitationally bound if their orbits are closed; gravitationally bound systems have a negative total mechanical energy Kepler’s first law law stating that every planet moves along an ellipse, with the Sun located at a focus of the ellipse Kepler’s second law law stating that a planet sweeps out equal areas in equal times, meaning it has a constant areal velocity Kepler’s third law law stating that the square of the period is proportional to the cube of the semi-major axis of the orbit neap tide low tide created when the Moon and the Sun form a right triangle with Earth neutron star most compact object known—outside of a black hole itself Newton’s law of gravitation every mass attracts every other mass with a force proportional to the product of their masses, inversely proportional to the square of the distance between them, and with direction along the line connecting the center of mass of each non-Euclidean geometry geometry of curved space, describing the relationships among angles and lines on the surface of a sphere, hyperboloid, etc. orbital period time required for a satellite to complete one orbit orbital speed speed of a satellite in a circular orbit; it can be also be used for the instantaneous speed for noncircular orbits in which the speed is not constant perihelion point of closest approach to the Sun of an orbiting body; the corresponding term for the Moon’s closest approach to Earth is the perigee principle of equivalence part of the general theory of relativity, it states that there no difference between free fall and being weightless, or a uniform gravitational field and uniform acceleration Schwarzschild radius critical radius (RS) such that if a mass were compressed to the extent that its radius becomes less than the Schwarzschild radius, then the mass will collapse to a singularity, and anything that passes inside that radius cannot escape space-time concept of space-time is that time is essentially another coordinate that is treated the same way as any individual spatial coordinate; in the equations that represent both special and general relativity, time appears in the same context as do the spatial coordinates spring tide high tide created when the Moon, the Sun, and Earth are along one line theory of general relativity Einstein’s theory for gravitation and accelerated reference frames; in this theory, gravitation is the result of mass and energy distorting the space-time around it; it is also often referred to as Einstein’s theory of gravity tidal force difference between the gravitational force at the center of a body and that at any other location on the body; the tidal force stretches the body universal gravitational constant constant representing the strength of the gravitational force, that is believed to be the same throughout the universe

Key Equations

 Newton’s law of gravitation $$\vec{F}_{12} = G \frac{m_{1} m_{2}}{r^{2}} \hat{r}_{12}$$ Acceleration due to gravity at the surface of Earth $$g = G \frac{M_{E}}{r^{2}}$$ Gravitational potential energy beyond Earth $$U = - \frac{GmM_{E}}{r}$$ Conservation of energy $$\frac{1}{2} mv_{1}^{2} - \frac{GMm}{r_{1}} = \frac{1}{2} mv_{2}^{2} - \frac{GMm}{r_{2}}$$ Escape velocity $$v_{esc} = \sqrt{\frac{2GM}{R}}$$ Orbital speed $$v_{orbit} = \sqrt{\frac{GM_{E}}{r}}$$ Orbital period $$T = 2 \pi \sqrt{\frac{r^{3}}{GM_{E}}}$$ Energy in circular orbit $$E = K + U = - \frac{GmM_{E}}{2r}$$ Conic sections $$\frac{\alpha}{r} = 1 + e \cos \theta$$ Kepler’s third law $$T^{2} = \frac{4 \pi^{2}}{GM} a^{3}$$ Schwarzschild radius $$R_{S} = \frac{2GM}{c^{2}}$$

Summary

13.1 Newton's Law of Universal Gravitation

• All masses attract one another with a gravitational force proportional to their masses and inversely proportional to the square of the distance between them.
• Spherically symmetrical masses can be treated as if all their mass were located at the center.
• Nonsymmetrical objects can be treated as if their mass were concentrated at their center of mass, provided their distance from other masses is large compared to their size.

13.2 Gravitation Near Earth's Surface

• The weight of an object is the gravitational attraction between Earth and the object.
• The gravitational field is represented as lines that indicate the direction of the gravitational force; the line spacing indicates the strength of the field.
• Apparent weight differs from actual weight due to the acceleration of the object.

13.3 Gravitational Potential Energy and Total Energy

• The acceleration due to gravity changes as we move away from Earth, and the expression for gravitational potential energy must reflect this change.
• The total energy of a system is the sum of kinetic and gravitational potential energy, and this total energy is conserved in orbital motion.
• Objects must have a minimum velocity, the escape velocity, to leave a planet and not return.
• Objects with total energy less than zero are bound; those with zero or greater are unbounded.

13.4 Satellite Orbits and Energy

• Orbital velocities are determined by the mass of the body being orbited and the distance from the center of that body, and not by the mass of a much smaller orbiting object.
• The period of the orbit is likewise independent of the orbiting object’s mass.
• Bodies of comparable masses orbit about their common center of mass and their velocities and periods should be determined from Newton’s second law and law of gravitation.

13.5 Kepler's Laws of Planetary Motion

• All orbital motion follows the path of a conic section. Bound or closed orbits are either a circle or an ellipse; unbounded or open orbits are either a parabola or a hyperbola.
• The areal velocity of any orbit is constant, a reflection of the conservation of angular momentum.
• The square of the period of an elliptical orbit is proportional to the cube of the semi-major axis of that orbit.

13.6 Tidal Forces

• Earth’s tides are caused by the difference in gravitational forces from the Moon and the Sun on the different sides of Earth
• Spring or neap (high) tides occur when Earth, the Moon, and the Sun are aligned, and neap or (low) tides occur when they form a right triangle.
• Tidal forces can create internal heating, changes in orbital motion, and even destruction of orbiting bodies.

13.7 Einstein's Theory of Gravity

• According to the theory of general relativity, gravity is the result of distortions in space-time created by mass and energy.
• The principle of equivalence states that that both mass and acceleration distort space-time and are indistinguishable in comparable circumstances.
• Black holes, the result of gravitational collapse, are singularities with an event horizon that is proportional to their mass.
• Evidence for the existence of black holes is still circumstantial, but the amount of that evidence is overwhelming.