Inertial and Gravitational Forces in Tennis
The tennis ball has a mind of its own. Or so it seems at times.
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The general nature of force, developed by Isaac Newton, is that it tends to accelerate objects, either positively or negatively, depending on their state. I.e., force tends to overcome inertia, the property of objects to resist changes in their velocity. Two sides of a coin, force is what overcomes inertia, and inertia is what is overcome by force.
The inertia of an object depends on its mass. The greater the mass, the greater the inertia and therefore the greater the force required to accelerate the object by a certain amount. In formal terms, force (F) equals mass (m) times acceleration (a):
F = ma.
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If you watch falling objects, you will see that heavy objects drop faster than light ones. A rock plummets, whereas a feather only floats down. In the Newtonian model of mechanics, this difference is due to air resistance, which tends to slow moving objects, some more than others. Feathers and tissue paper, for instance, show more resistance than stones or chunks of metal, proportionate to their weight, so they don't drop as fast.
But in a vacuum there are no air molecules to act against the falling objects, so feathers and paper drop just as fast as stones or chunks of metal. Gravity doesn't discriminate between different shapes, size, or weight.
Gravitational Acceleration
Since force equals mass times acceleration, acceleration, given by a, equals the force F, divided by the mass m,
The acceleration of an object for the particular force of gravity has to remain constant, because the numerator (the force) increases proportionately when the denominator (the mass) increases, and it decreases proportionately when the mass decreases.
For this situation, the gravitational equation takes the form:
F = (Gm1/r2)m.
So,
a = F/m = (Gm1/r2) = Constant.
This shows that the acceleration due to gravity is independent of the mass of the object attracted. Normally, we let the lower-case letter g stand for this special acceleration.
Force and Weight
The effect on bodies by the force of gravity is exactly what we know to be weight. You weigh a hundred and fifty pounds because you happen to be in the earth's gravitational field. On the moon you'd weigh less. Weight depends on the force of gravity, but it also depends on your mass. So when we say something is heavy, we could mean it's in a high gravitational field, or we could mean it has lots of mass. The weight, call it W, is:
W = mg
where m is the mass of the object and g is the acceleration produced by gravity on the object.
Following the same idea, the force or "weight" generated by steam power, say, or by your muscles on some object, is expressed with the same formula, except that we now use the symbols, F and g, instead of W and a. That is:
F = ma
where F is the force on the object, m is the object's mass, and a is the acceleration produced by the force on the object.
Force is the more general concept, because it could be applied in any direction in many different ways. Gravity, though, is strictly an attraction between objects.
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Every force can be resolved into any one of many paired, ninety-degree components. Each of the pairings can represent the force equally. Physically, this says the paired components can replace the original force and have the same effect.
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Inertial and gravitational forces both apply to the moving tennis ball. At any point in its trajectory, the ball is propelled by an inertial force, generated initially by a racket, and by gravity, which acts continually to pull the ball to the ground.
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