Force
According to Newton's second law, the rate of change of momentum of a body is proportional to the force acting on the body.
To see what this means, suppose a force F pushes a body of mass m. If there is no friction and no other forces, the velocity would increase with time. Since momentum is mass times velocity, so momentum would also increase. This momentum would increase at a certain amount every second. This is called the rate of change of the momentum. It can be calculated if we know the momenta at the start and end of a period of time. Then the rate of change is the final momentum mv, minus the initial momentum mu, divided by the time taken t. The rate of change of momentum can be written as a formula:
Newton's second law says that this is proportional to the force. This means that if I double the force, the rate would double. In this case, it means that over the same period of time, the momentum would change by twice as much. Such a relation can be written as
A constant is a number that stays the same even if the other quantities change. What is the value of this number?
One way to deal with this is to choose this constant to be 1. Then if the rate of change of momentum is also 1 (in SI units), we say that the force is 1 unit. This is how people usually do it today, and the unit of force is called Newton, or N.
The second law can then be written as
As an example, suppose a book of mass 0.2 kg is at rest on a smooth table. I push the book. After 10 s, the velocity of the book is 0.4 m/s. What is the force I applied?
Substituting into the above equation, we find
So far, I have assumed that there is a fixed body that accelerates as a force is applied. There is another possible situation. Imagine pushing an empty cardboard box on a smooth floor. As you push, I drop balls into the box one by one.
The mass of the box would increase. Suppose you apply just enough force to keep the box going at the same velocity. The momentum increases, this time because the mass increases, instead of the velocity. Once again, we can talk about the rate of change of momentum, and once again, this must be equal to the force.
For this situation, we could imagine an equation that looks like this:
As an example, suppose a box of mass 0.2 kg is at rest on a smooth floor. While I drop tennis balls into the box, you push it at a constant velocity of 0.1 m/s. After 10 s, the total mass is 1 kg. What is the force you applied?
Substituting into the above equation, we find
Copyright 2011 by Kai Hock. All rights reserved.
Last updated: 6 March 2011