(18) Newton's Second Law

--A force is the name given to whatever causes motion.
--The most familiar force is weight, the downward force on an object due to gravity. We can therefore measure force in grams or kilograms, units of weight, and loosely define force as "anything that can be matched by weight" (e.g. the tension of a spring).
--Forces can be opposed or unopposed.
--In the absence of opposing forces, if no force acts on an object at rest or moving at constant speed, it continues to do so indefinitely (Newton's first law).
--In the absence of opposing forces, if a force does act on an object at rest or moving at constant speed, it accelerates in the direction of the force.
--The acceleration of such an object is limited by its own resistance to motion, which Newton named its inertia.
--If air resistance can be ignored, a light object falls just as fast as one twice as heavy. Newton proposed that the reason was that although the force of gravity on the heavier object (its weight) was twice as large, so was its inertia.
In today's terms we say that both weight and inertia are proportional to the mass of the object, the amount of matter which is contains.

The MKS System and the "newton"

Consider free fall due to gravity. The force of gravity is proportional to mass m, so we can write

F = mg (1)
where g is the acceleration of gravity, directed downwards. Yes, proportionality allows one to add on the right some constant multiplier, but we won't, because now we want to define some units of F.

All quantitative formulas and units in physics depend on the units in which three basic quantities are measured--distance, mass. and time. Let us therefore choose from now on to measure distance in meters, mass in kilograms and time in seconds. That convention is known as the MKS system: as long as one's formulas contain only quantities derived by that system, they will be consistent and correct. But watch out... if by mistake you mix MKS units with grams or centimeters (or pounds and inches), and you may end up with some mighty strange results!

[This, indeed, was how the Mars Climate Orbiter--a 125 M$ space mission--was lost on 23 September 1999. When a small rocket was fired to adjust its entry to the Mars atmosphere, the operator, a NASA contractor, assumed its thrust was given in English units. Actually NASA specifications gave it in metric ones.]

In the MKS system the effective value of g varies from 9.78 m/s² on the equator to 9.83 m/s² at the poles, due to the Earth's rotation (see section #24a). Equation (1) not only shows that weight is proportional to mass, but--assuming it is measured in kilograms--it introduces a unit of F, named (no surprise!) the "newton."

By that equation, a force of one newton acting on one kilogram of mass accelerates it by 1 m/sec², so the force of gravity on one kilogram of mass is about 9.8 newtons. Earlier this was called "a force of one kilogram of weight, " a convenient unit for rough applications (1 kg = 9.8 newton), but not for accurate ones, because of the variation of g around the globe.

Newton's Second Law

We now can express in numbers the dependence of acceleration on force and mass. Lord Kelvin, leading British scientist in Queen Victoria's era, was quoted as once saying

"When you measure what you are speaking about and express it in numbers, you know something about it, but when you cannot express it in numbers your knowledge is of a meager and unsatisfactory kind... "

By Newton's second law, the acceleration a of an object is proportional to the force F acting on it and inversely proportional to its mass m. Expressing F in newtons we now get a--for any acceleration, not just for free fall--as

a = F/m (2)

One should note that both a and F have not only magnitudes but also directions--they both are vector quantities. Denoting vectors (in this section) by bold face lettering, Newton's second law should properly read

a = F/m (3)
This expresses the earlier statement "accelerates in the direction of the force."

Many textbooks write

F = ma (4)
but equation (3) is the form in which it is usually used--F and m are the inputs, a is the result. The example below should make it clear.

Example: the V-2 Rocket

The V-2 military rocket, used by Germany in 1945, weighed about 12 tons (12,000 kg) loaded with fuel and 3 tons (3,000 kg) empty. Its rocket engine created a thrust of 240,000 N (newtons). Approximating g as 10m/s², what was the acceleration of the V-2 (1) at launch (2) at burn-out, just before it ran out of fuel?

Solution Let the upwards direction be positive, the downwards direction negative: using this convention, we can work with numbers rather than vectors. At launch, two forces act on the rocket: a thrust of +240,000 N, and the weight of the loaded rocket, mg = –120,000 N (if the thrust were less than 120,000 N, the rocket would never lift off!). The total upwards force is therefore

F = + 240,000 N – 120,000 N = +120,000 N,
and the initial acceleration, by Newton's 2nd law, is

a = F/m = +120,000 N/12,000 kg = 10 m/s² = 1 g
The rocket thus starts rising with the same acceleration as a stone starts falling. As the fuel is used up, the mass m decreases but the force does not, so we expect a to grow larger. At burn-out, mg = –30,000 N and we have

F = + 240,000 N – 30,000 N = +210,000 N,
giving a = F/m = +210,000 N/3,000 kg = 70 m/s² = 7 g

The fact that acceleration increases as fuel is burned up is particularly important in manned spaceflight, when the "payload" includes living astronauts. The body of an astronaut given an acceleration of 7 g will experience a force up to 8 times its weight (gravity still contributes!), creating excesive stress (3-4 g is probably the limit without special suits). It is hard to control the thrust of a rocket, but a rocket with several stages can drop the first stage before a gets too big, and continue with a smaller engine. Or else, as with the space shuttle and the original Atlas rocket, some rocket engines are shut off or dropped, while others continue operating.

Next Stop: (18a) Newton's 3rd Law

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Author and Curator: Dr. David P. Stern
Mail to Dr.Stern: stargaze("at" symbol)phy6.org .

Last updated: 6.6.2004