4. Harmonic motion, gravitational field
In this lesson we would cover two topics: harmonic motion connected with spring-mass system and gravitational field.
Simple harmonic motion is governed by a restorative force. For a spring-mass system, such as a block attached to a spring, the spring force is responsible for the oscillation.
Since the restoring force is proportional to displacement from equilibrium, both the magnitude of the restoring force and the acceleration is the greatest at the maximum points of displacement. The negative sign tells us that the force and acceleration are in the opposite direction from displacement.
We can write from Newtonian Second Law that
, and also that for spring the force is equal
, where k is the spring constant, and x is displacement. By combining this two equation together we can achieve that
, and derived a is equal
.
We can graph the movement of an oscillating object as a function of time. Frequency f and period T are independent of amplitude A. We can find the period T by taking any two analogous points on the graph and calculating the time between them. It’s often easiest to measure the time between consecutive maximum or minimum points of displacement. Once the period is known, the frequency can be found using f=1/T.
Displacement as a function of time is proportional to amplitude and the cosine of . x is displacement as a function of time, where A is amplitude, f is frequency, and t is time, so we have
.
Nearly every child knows of the word gravity. Gravity is the name associated with the mishaps of the milk spilled from the breakfast table to the kitchen floor and the youngster who topples to the pavement as the grand finale of the first bicycle ride. Gravity is the name associated with the reason for "what goes up, must come down," whether it be the baseball hit in the neighborhood sandlot game or the child happily jumping on the backyard mini-trampoline. We all know of the word gravity - it is the thing that causes objects to fall to Earth. Yet the role of physics is to do more than to associate words with phenomenon. The role of physics is to explain phenomenon in terms of underlying principles. The goal is to explain phenomenon in terms of principles that are so universal that they are capable of explaining more than a single phenomenon but a wealth of phenomenon in a consistent manner. Thus, a student's conception of gravity must grow in sophistication to the point that it becomes more than a mere name associated with falling phenomenon. Gravity must be understood in terms of its cause, its source, and its far-reaching implications on the structure and the motion of the objects in the universe.
The gravitational force is relatively simple. It is always attractive, and it depends only on the masses involved and the distance between them. Stated in modern language, Newton’s universal law of gravitation states that every particle in the universe attracts every other particle 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.
The bodies we are dealing with tend to be large. To simplify the situation we assume that the body acts as if its entire mass is concentrated at one specific point called the center of mass. For two bodies having masses m and M with a distance r between their centers of mass, the equation for Newton’s universal law of gravitation is
,
where F is the magnitude of the gravitational force and G is a proportionality factor called the gravitational constant. G is a universal gravitational constant—that is, it is thought to be the same everywhere in the universe. It has been measured experimentally to be
.
The value of the attraction of gravity or of the potential is determined by the distribution of matter within Earth or some other celestial body. In turn, as seen above, the distribution of matter determines the shape of the surface on which the potential is constant. Measurements of gravity and the potential are thus essential both to geodesy, which is the study of the shape of Earth, and to geophysics, the study of its internal structure. For geodesy and global geophysics, it is best to measure the potential from the orbits of artificial satellites. Surface measurements of gravity are best for local geophysics, which deals with the structure of mountains and oceans and the search for minerals.
The acceleration g varies by about 1/2 of 1 percent with position on Earth’s surface, from about 9.78 metres per second per second at the Equator to approximately 9.83 metres per second per second at the poles.
The gravitational potential at the surface of Earth is due mainly to the mass and rotation of Earth, but there are also small contributions from the distant Sun and Moon. As Earth rotates, those small contributions at any one place vary with time, and so the local value of g varies slightly.
In contrast to the tremendous gravitational force near black holes is the apparent gravitational field experienced by astronauts orbiting Earth. What is the effect of “weightlessness” upon an astronaut who is in orbit for months? Or what about the effect of weightlessness upon plant growth? Weightlessness doesn’t mean that an astronaut is not being acted upon by the gravitational force. There is no “zero gravity” in an astronaut’s orbit. The term just means that the astronaut is in free-fall, accelerating with the acceleration due to gravity. If an elevator cable breaks, the passengers inside will be in free fall and will experience weightlessness. You can experience short periods of weightlessness in some rides in amusement parks.
Tidal forces are based on the gravitational attractive force. With regard to tidal forces on the Earth, the distance between two objects usually is more critical than their masses. Tidal generating forces vary inversely as the cube of the distance from the tide generating object. Gravitational attractive forces only vary inversely to the square of the distance between the objects. The effect of distance on tidal forces is seen in the relationship between the sun, the moon, and the Earth’s waters.
Kepler’s Law of Orbits – The Planets move around the sun in elliptical orbits with the sun at one of the focii.
Kepler’s Law of Areas – The line joining a planet to the Sun sweeps out equal areas in equal interval of time.
Kepler’s Law of Periods – The square of the time period of the planet is directly proportional to the cube of the semimajor axis of its orbit.