Download 7.4 – Universal Gravitation

7.4 – Universal Gravitation
Circular Motion
Velocity is a vector quantity, which means that it involves both speed (magnitude) and direction. Therefore an
object traveling at a constant speed can still accelerate if the direction is changed. The change in direction is
due to a net force on the object. This net force is called a centripetal force which results in centripetal
acceleration.
Centripetal means “toward the center.” Therefore both centripetal force, which results in circular motion, and
centripetal accelerate are both directed toward the center of the circular motion.
The reason that one feels pulled outwards while making a turn in a car is not due to a force but due to one’s
inertia. The inertia resists the change of motion and therefore the velocity of an object in circular motion is
tangent to the curve, or a vector coming straight off of the curvature as shown in the picture below of a mass
moving clockwise.
The equation for centripetal force is shown below:
𝑚𝑣 2
𝐶𝑒𝑛𝑡𝑟𝑖𝑝𝑒𝑡𝑎𝑙 𝐹𝑜𝑟𝑐𝑒 = 𝐹𝐶 =
𝑟
Example: A student has a 0.5 kg mass on the end of a 2 m string. The student twirls the mass in a
circular motion around their head at a constant speed of 4 m/s. What is the force of tension on the
string?
Answer:
𝑚𝑣 2
(0.5 𝑘𝑔)(4 𝑚/𝑠)2
𝐹𝐶 =
=
= 4𝑁
𝑟
2𝑚
7.4 – Universal Gravitation
Example: A car that has a mass of 1000 kg makes a turn at a radius of 30 m. If the force of friction
needed to keep that car on the road is 2700 N. What speed should the car travel in order to not slip off
the road?
Answer:
𝐹𝐶 =
𝑚𝑣 2
(1000 𝑘𝑔)(𝑣 2 )
=
= 2700 𝑁
𝑟
30 𝑚
𝑣2 =
(2700)(30)
= 81
1000
𝑣 = 9 𝑚/𝑠
Period (T) – The amount of time (in seconds) it takes for an object to make one full orbital rotation.
Frequency (f) – The amount of cycles an object makes in one second or Hertz (Hz).
𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 = 𝑓 =
1
𝑃𝑒𝑟𝑖𝑜𝑑 (𝑇)
Example: If a ball on a string takes 5 seconds to make one orbital period, what is its frequency in cycles
per second?
Answer:
𝑓=
1 1
𝑐𝑦𝑐𝑙𝑒𝑠
= = 0.20
𝑜𝑟 0.20 𝐻𝑧
𝑇 5
𝑠𝑒𝑐𝑜𝑛𝑑
7.4 – Universal Gravitation
Universal Gravitation
According to a popular legend, Isaac Newton discovered that gravity was a universal force, or a force that is not
just unique to Earth, while sitting under an apple tree on his mother’s farm. Newton understood Galileo’s
concepts of inertia that objects would continue to move in a constant speed in a straight line unless acted on by
a nonzero net force.
What Newton observed was that in the same way that the apple accelerated downwards due to a gravitational
force, the moon in a similar way does not follow a straight-line path, but instead circles the Earth. Therefore,
the moon which exhibited circular motion must in fact experience a net force. And so Newton reasoned that the
moon and the apple are both falling toward the Earth due to the Earth’s gravity.
And so Newton proposed that the moon was simply a projectile that was falling at the same curvature as the
Earth under the gravitational attraction of the Earth. And the moon is traveling at the exact speed (tangential
velocity) and is the correct distance (radius) away from the Earth so that it does not fall into the Earth or get cast
out into space.
Newton developed his Law of Universal Gravitation:
𝐹𝐺 = 𝐺
𝑚1 𝑚2
𝑟2
This law states that every object attracts every other object with a force called gravity. The force of gravity is
directly proportional to each of the masses (m1 and m2), and inversely proportional to the square of the distance
between them (r). According to Newton’s 3rd Law of Motion, the force of gravitational attraction is the same
for each of the two masses (as shown below). The G value in the equation is the universal gravitational
constant and is a constant of 6.67 x 10-11 N∙m2/kg2.
Example: What is the force of gravity of a 50 kg object that is on the surface of the Earth according to
Newton’s Law of Universal Gravitation?
(The Earth has a mass of 6 x 1024 kg and a radius of 6.4 x 106 m)
Answer:
𝐹𝐺 = 𝐺
𝑚1 𝑚2 (6.67 𝑥 10−11 )(50)(6 𝑥 1024 )
=
= 500 𝑁
𝑟2
(6.3 𝑥 106 )2
(Do not forget to use your E button on your calculator when putting in x10#)
7.4 – Universal Gravitation
Example: If a planet has double the mass and double the radius of Earth, what is the force of gravity on
that planet compared to Earth?
Answer:
𝐹𝐺 = 𝐺
𝑚1 𝑚2
(2)
2 1
=
=
=
𝑟2
(22 ) 4 2
The force of gravity on the planet would be ½ of that on Earth (g = 5 m/s2). The mass was doubled but the
radius was doubled to the second power or four.
Example: Which diagram below best represents the gravitational forces between a satellite, S, and
Earth?
Answer:
The third diagram (3) best represents the gravitational forces due to Newton’s 3rd Law of Motion and the Law
of Universal Gravitation.
7.4 – Universal Gravitation
Kepler’s Laws of Planetary Motion
Most of the scientific world in the 1600’s believed that the planets had circular orbits and the Earth was the
center of the solar system (geocentric model or Ptolemaic model of the solar system). An astronomer named
Johannes Kepler during that same time used data collected by his mentor Tycho Brahe, and developed three
laws governing the motion of planetary bodies in a sun-centered solar system (heliocentric model or Copernican
model of the solar system). These laws are still used today as an accurate description of the motion of any
planet and any satellite.
Kepler’s Three Laws of Planetary Motion are:
1. The orbits of planetary bodies are ellipses with the sun at one of the two foci of the ellipse. (Law of
Ellipses)
2. If you were to draw a line from the sun to the orbiting body, the body would sweep out equal areas in
equal intervals of time. (Law of Equal Areas)
7.4 – Universal Gravitation
3. The ratios of the squares of the periods of any two planets are equal to the ratio of the cubes of their
averages distances from the sun. (Law of Harmonies)
The orbital period is given in the units of earth years and the average distance is given in astronomical
units where 1 au is equal to the distance from the sun to the earth. Amazingly, every planet in our solar
system has the same period2/distance3 ratio. If one used exact numbers of orbital period and distance
from the sun, the ratio would still be equal to each other.
Example: What is the most accurate description of the shape of Mars’ orbit around the sun?
Answer: An ellipse
Example: Which planet takes the longest amount of time to make one complete revolution around the
sun - Venus, Earth, Jupiter, or Uranus?
Answer: Uranus is farthest from the sun, therefore according to Kepler’s 3rd Law of Planetary Motion it must
have the longest orbital period.
Example: Calculate the ratio of T2/R3 for each of Jupiter’s moons listed in the table below. Indicate
whether this confirms or contradicts Kepler’s 3rd Law of Planetary Motion.
Answer: a. 3.16 x 10-16, b. 3.13 x 10-16, c. 2.87 x 10-16, and d. 3.02 x 10-16. All of these ratios are
approximately the same and it confirms Kepler’s 3rd Law of Planetary Motion.
7.4 – Universal Gravitation
Phases, Eclipses, and Tides
Motions of the Moon
The moon revolves around Earth and also rotates on its own axis. As the relative positions of the moon, Earth,
and sun change, it causes the phases of the moon, eclipses, and tides.
The moon rotates once on its axis in the same amount of time as it revolves around the Earth, therefore a “day”
and a “year” on the moon are approximately the same length. This is the reason why the same side of the moon
always faces Earth.
Phases of the Moon
The moon does not produce light, but simply reflects the light from the sun. The different shapes of the moon
visible from Earth are called phases. The moon goes through its whole set of phases each time it makes a
complete revolution around Earth. These phases are caused by the relative positions of the moon, Earth, and the
sun. The phase of the moon you see depends on how much of the sunlit side of the moon faces the Earth.
The phases of the moon are as follows:








New Moon – The sun (and therefore the sunlit side of the moon) is behind the moon.
Waxing Crescent – The portion of the observable moon is waxing, or growing into a crescent shape.
First Quarter – Half of the sunlit side of the moon is observed.
Waxing Gibbous – As the moon continues to wax, or grow, the visible side is called gibbous.
Full Moon – The entire sunlit side faces the Earth.
Waning Gibbous – The visible side is waning or shrinking.
Third Quarter – Half of the sunlit side of the moon is observed.
Waning Crescent – Again a crescent is observed, but it is waning.
7.4 – Universal Gravitation
Eclipses
The moon’s orbit around the Earth is slightly tilted (about 5º) with respect to Earth’s orbit around the sun.
Therefore, most months neither the Earth’s shadow nor the moon’s shadow affects one another. However, an
ellipse occurs when the moon’s shadow is cast onto the Earth or the Earth’s shadow is cast onto the moon.
The two types of eclipses are a solar eclipse and lunar eclipse.
During a new moon, the moon lies between the Earth and the sun and usually the moon travels a little above or
below the sun it in the sky. A solar eclipse occurs when the moon passes directly between the sun and the
Earth, blocking sunlight from the Earth and casting a shadow on the Earth.
The moon’s shadow during a solar eclipse is called the umbra, and it is a very dark, cone-shaped part which
blocks the suns completely. During this time on Earth, people within the umbra experience a total solar eclipse
and the sky is as black as night. Stars are visible in this total solar eclipse as well as the solar corona, the outer
atmosphere of the sun. The moon also casts another part of its shadow during this time in a larger part called
the penumbra. This is only a partial eclipse and since an extremely bright part of the sun remains visible, it is
not safe to look directly at the sun during this partial solar eclipse.
A lunar eclipse occurs at a full moon when the Earth is directly between the moon and the sun. The moon is
then in the Earth’s shadow and it looks dim from Earth. Similar to a solar eclipse, the Earth’s shadow has an
umbra and a penumbra. When the moon is in the Earth’s umbra, you can see a total lunar eclipse. Unlike the
precise nature of the solar eclipse, a lunar eclipse can be observed from anywhere on Erath that the moon is
visible; therefore it is more likely to observe a lunar eclipse than a solar eclipse.
7.4 – Universal Gravitation
Tides
The rise and fall of ocean water which is called tides occur every 12.5 hours (about six hours of rising and then
about six hours of falling). This is caused by the combined effects of the gravitational forces exerted by the
moon, sun, and rotation of the Earth.
High tide occurs primarily when the moon’s gravity is strongest due to the closer proximity to the moon. High
tide also occurs on the far side of Earth from the moon. Low tides occur between the two high tides.
7.4 – Universal Gravitation
When the gravitational effects of the sun and the moon pull in the same direction (at a new moon and a full
moon), their combined forces produce what is called a spring tide. During the moon’s first and third-quarter
phases, the sun’s pull and moon’s pull are at right angles and produce a neap tide, which produce very little
difference between low and high tides. Both spring tides and neap tides occur twice a month.