Module 2 -- Newton's Second Law
From PER wiki
Newton's Second Law describes how the motion of an object changes as a result of all interactions it is experiencing.
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.
|Jump over the Great Wall|
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.
|Review: motion and change in motion in straight line|
|To move means that the position changes as time "ticks" on. Both velocity and acceleration characterize the motion; but they describe two very different aspects of motion.
Velocity is defined as the rate of change of position. In one dimension, or straight line motion then vx = dx / dt. Alternately the velocity at a particular moment can be represented by the slope of the graph of the objects position vs. time, evaluated at the particular moment of interest.
This is the second derivative of position or the familiar a = d2x / dt2 = dvx / dt. Considering the graphical representation again, the acceleration of an object moving in the x-direction can be determined by finding the slope of the velocity vs. time graph at the moment of interest.
Newton's Second Law is traditionally written as:
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,, 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
When applying Newton's second law in problem solving the most important aspects of Newton's second law to remember are:
|Newton's second law is a LAW OF CHANGE:|
|Only a non-zero net force will result in the object's acceleration, which is a change in velocity, hence a change in the state of the object's motion. The few expressions that tell us how a physical quantity changes with time are very useful for solving problems as well as leading to understanding of the underlying physical state of a system. In this course we are particularly interested in LAWS OF CHANGE because of this.
Newton's second law is a physical definition of acceleration; the detailed motion definition will be examined in detail later in the course. For now, all you need to know is that acceleration is defined by the above relationship, the net force scaled in inverse proportion to the mass. Standard units are m/s2. It is a Law of Nature that objects which are changing velocity are responding to a net force acting on them, this is why Newton's Second Law is the LAW of CHANGE:
On the left hand side: the derivative means the rate of change of the velocity over time (change in motion)
NOTICE: Because this is a vector relationship then the change of the velocity (not the velocity itself) has the same direction as the net force.
|is NOT a force! Physics is not equal to Mathematics!|
| This is an example of where physics and mathematics diverge. Newton's Second Law is an equality between two "different" quantities: and are two DIFFERENT quantities.
On the left hand side: force is the quantitative resulting force from all interaction(s) with other objects (EXTERNAL to the object). We call it "net" force to indicate the importance of ALL the interactions. The "net force" is not "one of the forces" acting on the object. Rather, the net force can be thought of as the single force that could be applied to the object to cause the same consequence as the effect of all the forces actually acting on the object.
On the right hand side : m is the mass (an intrinsic property of the object understood in terms of Newton's First Law), and is the acceleration of the object, describing how the motion is changing. The product of these two physical quantities is equivalent to the net force acting on the object. This does not mean that is a force of any kind, nor an interaction of any kind with the object. is not a force!
|Note to students who hunt for equations as a problem solving strategy!|
|One of the languages of physics is mathematical expressions as in Newton's Second Law. However, students who inquire only as far as identifying the equation that represents the ideas of this law, frequently misunderstand the interpretation of the right hand side of the equation and make numerous mistakes in problem solving. Therefore, it is a good intellectual practice to restate in your own way the physical meaning of each term in any basic equation in physics. Write it down, think about it. And compare your ideas with your understanding of more basic ones to develop the skill of checking your newer ideas for consistency with those well understood ideas that you already claim.|
|Force units are or Newtons (N)|
|Force is measured in Newtons (N) (in the International System of Units.) Newton's Second Law provides an operational (or procedural) definition for the unit of force. A net pushing force of 1 N is the magnitude of the force that will cause a mass of 1kg to accelerate with an acceleration of 1 m/s2. Dimensionally, force is equal in magnitude to the unit of [mass] . [acceleration] which becomes .
Therefore 1 N is equivalent to 1 kg m/s2 .
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.
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
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
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.