### Re: Does Newton's Second Law apply to all forces?

Date: Fri May 26 15:24:25 2000
Posted By: Tom Cull, Staff, Clinical Sciences MR Division, Picker International
Area of science: Physics
ID: 958313672.Ph
Message:

Hi Larry,

Newton's Second Law is a summary. The underlying principle it represents works in all cases. I found a really slick looking page about Newton's Laws that explains the concepts. Beyond the often quoted wording is a enormous world of mathematics, physics, and yes, even philosophy. Best of all Newton's Laws work all the time if some modernization due to new understanding is considered.

Newton's Second Law states: The acceleration produced by a net force on a body is directly proportional to
the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass
of the body.

Or in quick notation, this is often written Force = mass * acceleration, (see the webpage above)
where it is understood that force and acceleration are vectors and mass is a scalar.

Note that the law states net force. The equation should be applied for the sum of forces exerted. In the case where net force is zero then acceleration is zero.

Newton's 2nd Law in mathematical form is best expressed in the differential form:

d(momentum)/dt = d (mass * velocity)/dt where momentum and velocity are vectors.

This differential form is often not understood by folks who have no experience with differential equations and is often forgotten by people who do know. It is a form of the conservation of momentum. If d(momentum)/dt = 0 then momentum is constant (conserved).

The full differential has two terms because the mass could change as well. Performing the differential (chain rule)

d(momentum)/dt = velocity * d(mass)/dt + mass * d(velocity)/dt.

This equation is rarely considered in this form because mass is often constant. If mass is constant, we arrive at Newton's 2nd Law:

d(momentum)/dt = mass * d(velocity)/dt.

in which you recognize d(velocity)/dt as acceleration.

This equation is true for the 4-vector momentum of special relativity. Where momentum has modified definition and the fourth element of the vector is related to the energy of the object.

Similarly it works for angular momentum.

d(angular momentum)/dt = d (distance from torque point X velocity * mass)/dt

where momentum, distance from torque point and velocity are vectors. Also the X represents the vector cross product.

The differential form also works for photons which are massless. The difference is that momentum is defined differently,

photon momentum = Planck's constant / wave length.

Let's take another look at your example.
First, the units do not match up. Pounds are measure of force. But I think what you are trying to say is that 100 pounds of force can move 200 pounds and you are correct. So what other forces are at work?
Suppose we put the 200 pounds of weight on scale and attach a rope to it that I am going to pull to try to lift the weight. If I pull the rope straight up with a force of 100 pounds the weight does not lift, but the scale upon which it sits will read only 100 pounds. Similarly if I pull with 150 pounds of force the scale will only read 50 pounds of weight. The scale is reading the net force is the downward direction. Only when I am able to generate 200 pounds of force or more will the scale read zero and the weight will move upward.

Remember, Newton's 2nd Law is net force, plain, simple, and tacidly eloquent.
But more importantly, the law really represents a glimpse into a much richer realm of fundamental principles that have helped people understand how the world around them and the world beyond their common experience works.

Sincerely,