Module 2 -- Newton's Second Law

Newton's Second Law describes how the motion of an object changes as a result of all interactions it is experiencing.

Learning Goals

After completing this module you should be able to:

• Give two qualitatively different definitions of net force.
• State Newton's Second Law.
• State the most general conditions for Newton's First Law to apply to the motion of an object.
• Given a free body diagram for an object, write the vector equations of Newton's Second Law for that object.

Jumping the Great Wall of China

If you observe Danny Way's jump over the Great Wall of China, you will be able to identify at least five different segments along his path where his motion is clearly different: speeding up on the downhill curve, a horizontal stretch, coming up a short hill, flying over the wall and finally, traveling along a straight downhill ramp. All of this motion is viewed with respect to the fixed ground. In the design of the ramps, applying Newton's Second law was critical.

2nd Law Specifies Acceleration

Newton's 2nd Law does not state what is required for something to be in motion; the 1st Law addresses that. The 2nd Law specifically quantifies the rate of change of motion (acceleration) any object will undergo as a consequence of all of the forces on it. Because the law is describing changes in translational motion, any path deviating from a straight line occurs because of some net force; any other change in velocity also occurs because of some net force. The actual size of the acceleration depends on internal properties of the object and the external interactions imposed upon it.

Mathematical Form

Newton's Second Law is traditionally written as:

$\vec{F}_{net} = m\vec{a}$

The net force applied on an object causes an acceleration that is inversely proportional to the mass of the object. Larger mass makes objects more difficult to turn or change speed. The net force,$\vec{F}_{net}$, is the resulting force obtained by adding all the forces applied on the object. For an object having mass m that has three different forces acting on it, then $\vec{F}_{net} = \sum_{n=1}^3 \vec{F}_{n} = m\vec{a}$

When applying Newton's second law in problem solving the most important aspects of Newton's second law to remember are:

Equilibrium : Acceleration is zero.

What evidence is there of interaction forces when there is no change in motion? If we observe accelerations of some or all parts of a system (or object), then we infer the presence of unbalanced forces (Newton's 2nd), and conclude the presence of forces. Deformation of a rubber ball when it hits the floor is evidence of a force on the rubber ball (by the floor); similarly, the change in velocity of the ball when it hits the floor is evidence of this same force acting on the ball. A trickier example to unpack is an object at rest, where we observe zero acceleration.

When the net force on an object is zero, the object is said to be in equilibrium, meaning that the translational motion will not change. In this case, Newton's First Law applies.

We can invert this argument: if an object is moving with constant velocity (or, as a special case, if an object is at rest) then the forces acting on that object must add up to zero. This is the first place where the consequence of the vector nature of forces is trully evident.

Equilibrium Condition in Vector Notation

$\vec{F}_{net} = \sum \vec{F} = 0$

It is important to remember that force is a vector, which is what the arrows above the F symbols in the equilibrium equation denote. Vectors are a mathematical way to account for quantities that have three distinct components in 3 dimensional space (x,y,z). Vectors are quantities that have relative magnitude (size) AND point in some particular direction. Because of this, the equilibrium condition that the net force be zero is actually three separate identical conditions, one for each direction in space. Each direction can be considered alone, so three equilibrium equations must hold simultaneously. For each direction the sum of the components of the forces in that direction must equal zero. (Recall, having multiple simultaneously true equations allows for solving problems with multiple unknowns. Recognizing that one vector equation is equivalent to multiple simultaneous equations is very useful.)

Thankfully, the concise way to express the condition for equilibrium is:

Equilibrium Condition in Component Notation

$F_{net,x} = \sum F_{x} = 0$

and

$F_{net,y} = \sum F_{y} = 0$

and

$F_{net,z} = \sum F_{z} = 0$

Because two directions are sufficient for understanding most of translational mechanics, our examples and discussions will primarily focus on 1D and 2D motion. We will add the third dimension when it is absolutely necessary to understand the physics.