Fu = G*m2*m/Ru2 Fl = G*m2*m/Rl2 m = (mass of man)/2 = 50kg G = 6.67x10-11Nm2/kg2 |
Suppose we consider a man with a total mass of 100kg. We will calculate
the total force due to gravity on this man exerted by a body of mass m2.
We suppose that the upper half of the man is at a distance Ru from m2,
and the lower half is at a distance Rl from m2. Suppose the upper half
of the man has a mass of 50kg and likewise the lower half. Then the total
force, Ftot is
Ftot = Fu + Fl, and the tidal force is Ftidal = Fu - Fl |
The effect of the gravitational force is two fold. In the first case Ftot causes the two masses to accelerate towards each other. The second effect of the gravitational force is a stretching of the bodies because not all parts of the man, in this case, are equally distant from m2, and each part is attracted to m2 with a force that depends on the distance from m2 of that part. This second, stretching force is called the tidal force. On earth it gives rise to the familiar ocean tides. The ocean tides are caused by tidal forces due to the moon and sun. The tidal force depends more strongly on distance than the total force. For the case where the distances are nearly equal, that is to say
(Rl - Ru) << (Rl + Ru)/2 then
Ftidal = 2*(G*m2*m/R2)*(Rl-Ru)/R = Ftot*(Rl-Ru)/R
where R is the average distance , R = (Rl+Ru)/2.
This means that a small mass m2 could still have a large tidal effect even if its total force is small, provided its distance is small.
Although the total force basically determines the size
of orbits, the tidal forces can also have an important effect in the long
term because of the physical principle of the "conservation of angular
momentum". This principle states that quantity of rotational momentum
in a system of particles is constant in time if the system is not disturbed
by outside forces ( actually torques).
This principle is what enables us to ride a two wheel
bicycle. You can also see this effect in an ice skater's speed of
rotation as she draws her arms closer to her body. Her speed of rotation
may change, but the total angular momentum remains the same.
For our earth-moon system tidal forces have caused long term pronounced features in our joint motion.
i) The ocean tides, through friction, cause the earth's spin to slow down so that by the end of a year the length of the day has increased by 50x10-8 seconds, or 33 seconds/century. At the same time the moon's distance from the earth changes so as to keep the total angular momentum constant.
ii) Tidal forces of the earth on the moon have caused the moon's rate of rotation on its axis to become equal to its rate of orbital rotation, thus the moon always keeps the same face towards us.
iii) In the early days of the earth-moon system it is believed that the moon was much closer to the earth than at present. The period of the moon's orbital motion was closer to 7days, instead of the present day value of 27.5 days. The solar day was 6 hrs. This would produce tremendous ocean tides.
There are many other effects of tidal forces evident
in the solar system. For example, the period of mercury's axial rotation
is closely related to its orbital motion( in the ratio of 2/3). The tidal
effects of Jupiter, say, on its moons causes enough internal generation
of heat ( due to deformations) in the moons' interior to make some of the
moons volcanically active (Io).
Exercise
Some Solar system statistics and effects on our 100kg
man. Assume Rl-Ru=1m.
planet | m(kg)x1024 | distancex109m from sun for planets | Ftot (N) for closest approach | Ftidal (N) for closest approach |
mercury | 0.323 | 57.9 | ||
venus | 4.87 | 108 | ||
earth | 5.98 | 150, radius of the earth=6.38x106m | ||
mars | 0.645 | 228 | ||
jupiter | 1900. | 778 | ||
saturn | 569. | 1426 | ||
uranus | 86.7 | 2868 | ||
neptune | 103. | 4494 | ||
pluto | 0.36? | 5900 | ||
sun | 1.989x1030 | |||
moon | 0.0735 | 0.384 from earth | ||
man | 100kg | 1 |
Calculate the total force and the tidal force on our
100 kg man, who is standing on the surface of the earth, due to
1) the earth
2) the sun
3) the moon
4) jupiter
5) another 100 kg man standing 1m away.
Based on your results how signifcant, physically, is the position of the planets on our 100kg man ?