Friday, July 13, 2018
Unizor - Physics4Teens - Mechanics - Dynamics - Weight
Notes to a video lecture on http://www.unizor.com
Weight of an object, by definition, is the force of gravity a
planet attracts this object with. Usually, the word "weight" implies the
magnitude of this force; its direction is, obviously, always towards a
center of a planet.
So, weight is not a characteristic of an object itself, it's a
characteristic of an object relative to a planet. In most cases, this
planet is our Earth, though we sometimes say, for example, that a
particular object weighs on the Earth 6 times more than on the Moon.
This only means that the force of gravity on the surface of the Moon is 6
times weaker than on the surface of the Earth.
Do we feel weight as the force of gravity?
Not quite. What we can feel is pressure (reaction force) from the surface we stand on, that equalizes gravitational force to hold us at fixed position on a floor or on a ground.
If there is no support (like for a person jumping with a parachute from
an airplane before a parachute is open, if we ignore the air
resistance), we don't feel weight, we are weightless. We have different
senses, but not a sense of gravity.
So, feeling weightless is not really an absence of gravity, it's absence
of a reaction force that balances the gravity (equal in magnitude and
opposite in direction) and holds us fixed relatively to a planet.
This reaction force is not just against our feet, when we stand on the
floor, it's everywhere inside our body as well, since the body maintains
its shape. We feel this pressure of a reaction force everywhere inside.
That's why it's very difficult to emulate the gravity with some special
equipment on a spaceship.
People on a spaceship with non-working engines flying around the Earth
on an orbit feel weightless, because they are constantly falling towards
the Earth together with a spaceship (no support!) from the straight
line trajectory tangential to an orbit; planet attracts them with
gravitational force, and only because of the speed, they maintain
constant distance from the planet.
Since weight is a force, it is measured in units of force, like newtons in SI.
The weight of an object of mass m on a surface of a planet of mass M and radius R is, as we know,
W = G·M·m /R²
where G is a universal gravitational constant,
G = 6.674·10−11 N·m²/kg²
Since we are talking about weight as a force, which is a subject to the
Newton's Second Law, we can determine the acceleration this force causes
to an object of mass m, if acts alone:
a = W/m = G·M /R²
Notice that on the surface of Earth this acceleration is constant since
all components of this expression (gravitational constant G,
mass of Earth M and its radius R) are constants.
So, we can calculate this constant once and for all and, knowing the mass of an object m,
we can determine its weight by multiplying it by this constant, which
is, as we determined in the previous lecture, an acceleration of free
fall, which on the surface of Earth is traditionally symbolized by
g = G·M /R²
The value of this constant is, approximately, 9.8 m/sec².
But, to be precise, it's not the same at different points on the Earth
because the shape of the Earth is not exactly a sphere and its mass is
not uniformly distributed within its volume.
Moreover, it obviously changes with height (getting smaller) since the higher elevation is equivalent to a greater radius R (distance to a center of the Earth) of an object.
Now we can say that for an object of mass m the weight on the surface of the Earth is W=m·g=9.8·m. If mass m is measured in kilograms, this weight is measured in newtons.
Analogous calculation for other planets, based on their mass and radius, show the following values of free falling acceleration:
on Sun - 274.1 m/sec²
(objects are 28 times heavier on Sun than on Earth),
on Jupiter - 25.93 m/sec²
(objects are about 2.6 times heavier on Jupiter than on Earth),
on Moon - 1.625 m/sec²
(objects are about 6 times lighter on Moon than on Earth).
Historically, the weight is rarely measured in newtons. More customary units are:
1 pound (abbreviated lb) equals to 4.44822 newtons - the weight of an object of mass 0.454 kg on Earth;
1 kilogram-force (usually, simply called 1 kilogram, skipping "-force", and abbreviated kgf, but plain kg can also be used, when implication to weight is obvious) equals to weight of an object of mass of 1 kg on Earth, that is 9.8 newtons;