Chapter 9: Force

9.2 Newton’s laws

In the late 1600’s and early 1700’s, Isaac Newton published the Principia Mathematica in which he built upon discoveries of scientists (at the time called “natural philosophers”) before him, laying the foundation of classical mechanics. In the Principia, he described three laws of motion:

1. An object’s momentum will only change if the object is acted on by some force. This is known as the principle of inertia.
2. The total force acting on an object is equal to the rate of change of the object’s momentum:

   F_net = ^Δp⁄_Δt

3. For every force acting on an object, there is a reaction force equal in magnitude and opposite in direction acting on another object.

We have already discussed the first law in the context of energy; if you can understand inertia, you can explain many phenomena you observe. The second law is a mathematical statement which we can use to quantitatively solve problems. The third law, as we’ll see in the following example, is a result of conservation of momentum.

Example

Note: in the previous example, in the last line of math before the final answer, each velocity should be multiplied by its respective mass. The calculated result is correct, however.

Remember: all forces describe interactions between two objects; all forces come in pairs.

Finally, let’s take another look at the second law. Consider a situation where an object’s mass remains constant (which is often the case). The change in momentum would be:

Δp = p_f – p_i = mv_f – mv_i

Since the mass is not changing, we can factor it out:

Δp = m(v_f – mv_i) = m Δv

And substituting this into Newton’s second law gives us

F_net = ^Δp⁄_Δt = m ^Δv⁄_Δt

but ^Δv⁄_Δt is the object’s acceleration! So, when mass is constant, Newton’s second law can be written as

F_net = ma

Remember that force, momentum, and acceleration are all vector quantities. To work with them, we must separate into x and y components:

F_{net, x} = ^Δp_x⁄_Δt = m a_x

and

F_{net, y} = ^Δp_y⁄_Δt = m a_y

The SI unit of force is named the newton (N) in honor of Isaac Newton. Using Newton’s second law, we can break the newton down into base SI units:

N = kg · m/s²

For comparison, one newton is approximately the force required to push down one key on a computer keyboard.

Example

Practice

Practice

Practice