I.Newtonian Mechanics
Lesson 2: Part 2
Netwon’s Laws
I.Newtonian Mechanics
A.The Equation
In Newtonian Mechanics, the motion of a particle is described by Newton’s 2nd Law (A 2nd Order Differential Equation):
The notation which the book uses is simpler than the more explicit form that you should have seen in an introductory physics course. The vector signs and summation sign are implied!! This leaves a greater burden upon you as a reader to properly interpret the notation.
One should note that the time derivatives are full derivatives and not partial derivatives. Also the fact that Newton II is a 2nd order differential equation explains why this type of equation has been so important in the history of the development of mathematics.
B.Applying Newton 2nd Law
In theory, the solution of any classical mechanics problem could be accomplished by pursuing the following simple process:
1)Choose a coordinate system
2)Determine the net external force upon the particle (This usually involves drawing a free body diagram and an inventory of forces)
3)Use Newton’s 2nd Law to find the particles acceleration
4)Integrate the acceleration and use the a knowledge of the particle’s velocity at one instant in time to find the particle’s velocity at any instant of time.
5)Integrate the velocity and use the a knowledge of the particle’s position at one instant in time to find the particle’s position at any instant of time.
The reality is that this solution technique is rarely possible due to the mathematical difficulties that may arise from different types of forces.
C.Some Difficulties With The Straight Forward Solution Method
1)The forces vary as a function in space in such a way as to make the free body diagram vary!!
EXAMPLE: Consider the simple example of a block sliding down the wooden bowl shown below.
If we draw the free body diagrams at three different points (A,B, and C), we see that the free body diagrams are all different. This means that the force side of Newton 2nd Law for each dimension is also changing as we move from A to B to C. This change is due to the constrained motion of the particle and we may or may not be able to easily write this connection (i.e. parameterize the forces) in such a way as to handle the problem directly.
2)Forces which vary with velocity (drag, etc)
You need the forces to find the acceleration so that you can find the velocity. However, you can’t find the acceleration without knowing the velocity. There are differential equation solution techniques for handling this problem in simple cases like those in Chapter 2.
3)The forces create a differential equation which is extremely difficult to solve or the integrals have no closed forms.
Advanced differential equation solution techniques including analytical methods must be applied.
These are only some of the problems for the simplest possible
problem (only 1 slow moving particle). Real problems may involve
large numbers of particles whose interactions depend on the location
of the particles (solving multiple couple differential equations), a finite object whose mass might change (erosion of a missile in flight), interactions might occur so fast that the forces change drastically over short time intervals and can’t be measured (collisions & explosions in which the functional form of the forces is unknown).
I am not suggesting yet that Newton’s 2nd Law is flawed (It is due to
its reliance on absolute motion and experimental evidence for
deviations at high speeds). I am suggesting that the mathematical
difficulties make a frontal assault an un-winnable proposition in many
cases.
D.Solution
We have developed several solution techniques.
1) We have learned to categorize Newton’s 2nd Law based upon the forces and math difficulties and to use different mathematical techniques to solve different categories of problems.
2) We have developed new concepts and searched for physical quantities which remain unchanged in the problem (energy, linear momentum, etc). This has led us to develop the conservation laws which turn out to be more fundamental than Newton’s 2nd Law!!
3) We have developed alternative formulations of classical mechanics often based upon scalar quantities (Hamiltonian Mechanics, Lagrangian Mechanics, Principle of Virtual Work, etc).
Thus, part of the task of learning classical mechanics is to be able to recognize the various categories of problems (harmonic motion, constant acceleration, collisions, central force, etc ), the math technique or mechanics formulation that should be used to solve each category of problem and to have some knowledge of standard results obtained for the most simple problems in each class of problems.
II.Constant Net External Force Problems
1.Fact - A particle that is acted upon by a constant NET External
Force will undergo constant acceleration.
2.Solution Technique – Direct Application of Newton’s 2nd and Integration.
3.Equation of Motion & Its Solution (Kinematic Equations)
“Equation of Motion”
We integrate both sides with respect to time in order to find the velocity.
“Definition of Anti-Derivative”
OR“1st Result”
We integrate both sides with respect to time in order to find the position.
“Definition of Anti-Derivative”
OR“2nd”
We can obtain a third equation in which time is eliminated by using the definition of average velocity from physics and statistics.
The average velocity is defined in physics as the ratio of the particle’s displacement over time.
Using the definition of a statistical average (mean), we have
For the case of constant acceleration, the integral is simply the sum of the areas of a triangle and rectangle as shown in the diagram.
From the drawing, we see that for constant acceleration the integral is
Thus, the average velocity is
Note: This result depended upon the v-t graph so it is only true for a constant acceleration (i.e. straight line graph).
Combining our statistical result with the physics definition of average velocity, we have that
We then re-arrange the formula in terms of time as
We now substitute this into our velocity results and get
or “3rd”
The three equations that we have developed are called the kinematic equations and allow us to solve any constant acceleration problem. They are not the solution to Newton’s 2nd Law for a particle undergoing non-constant acceleration!!! Remember that a vector like acceleration is constant only if both its magnitude and direction remain unchanged or if equivalently all of its components are constant!!
III.Definition of Work
The infinitesimal amount of work done by a force, upon a particle as the particle is displaced an infinitesimal amount is defined as
If the particle is moved from one location to a second location then
the total work done by the force upon the particle is
The value for this integral (the amount of work done) usually depends on the path used in moving the particle. If the work done by a force depends only upon the initial and final locations of the particle and not the path taken by the particle then the force is called conservative. Otherwise the force is said to be non-conservative.
IV.Conservative Forces & Potential Energy
If a force depends only on the spatial coordinates such that
“Test To See If F Is Cons.”
then the force is a conservative force. These forces are important because we can replace a conservative force with the negative gradient of a scalar function called its potential energy function.
“Def. Of Potential Energy”
This replacement is not confined to classical mechanics. It is actually just an application of a fundamental fact from vector calculus that the curl of the gradient of any scalar function is always zero!! In mechanics, the vector is a conservative force and the scalar function is called the potential energy function. In electrostatics, the vector is the conservative electrostatic field and the scalar function is the electric potential (or voltage).
From vector calculus, we also know that the change in a scalar
function, U, that is a function of several variables is found by
Since our forces are time independent, the potential energy function
also doesn’t depend on time. Thus, we have
We can rewrite the equation in Cartesian coordinates as
If we now integrate this equation, we can obtain the change in the
potential energy function
Substituting into the right hand side our potential energy-conservative
force relationship, we see that
The left-hand side is the change in the potential energy while the right
hand side is the negative of the work done by an external conservative
force upon the particle.
If a conservative force does negative work upon a particle, it is
transferring potential energy to the particle.
V.Newton’s 2nd Law and Work (Work Energy Theorem)
If the work done by a conservative force decreases the potential
energy of a particle, what does the work by the net external force do? (i.e. What is the effect of all the work done by forces upon the particle?)
Using Newton’s 2nd Law to replace the net external force, we have
We now use the chain rule from Calculus and the definition of velocity to rewrite the last integral.
We can now use the definition of an anti-derivative to evaluate the
right hand side.
Thus, the work done by the net external force is equal to a change in a
scalar quantity which depends on the change in the speed of the
particle. The scalar quantity is called kinetic energy.
This extremely important equation is Newton’s 2nd Law in Energy
form and is called the “Work-Energy Theorem.”
The effect of the work done by all of the external forces acting on a
body is to change the kinetic energy of the body!!
Kinetic energy is defined as the energy that a body has due to its
motion. For a particle (i.e. a system with constant mass), kinetic
energy can be calculated using
VII.Total Mechanical Energy
If we divide the forces acting on a particle into those forces that are
conservative and those force that are non conservative, the Work-Energy Theorem (Newton’s 2nd Law in terms of energy) becomes
However, the last term is the change in the potential energy of the body so
The left hand side is the work by non-conservative forces while the
right hand side is the difference in a scalar quantity that consists of the
sum of the kinetic and potential energy. We call this new scalar
quantity the total mechanical energy of the particle.
E = K + U
The effect of work by non-conservative forces upon a particle is to
change the mechanical energy of the particle.
VIII.Conservation of Mechanical Energy
In some problems the forces only depend on the relative position of the particles so the forces are conservative. In this case, the work by non-conservative forces is zero and the mechanical energy of the particle is conserved.
We can rearrange this condition to solve for the kinetic energy
Using the kinetic energy formula for slow moving particles, we have
Since the speed is the magnitude of the velocity vector it must be positive. A classical particle is limited to regions in space where its potential energy is less than or equal to its mechanical energy since the particle can’t have an imaginary speed. A helpful way of looking at this type of problem is to graph the potential energy vs position as shown in Figure 2.3.1 (pg. 51) of your textbook by Fowles & Cassidy. For each location in space (x-value in the graph), the distance below the dashed energy line and the solid potential energy curve corresponds to the particle’s kinetic energy.
For a 1-D problem, the velocity of the particle is either plus or minus the speed depending on the direction that the particle is traveling (+ ot – x-direction). Thus, we have
Since each term in the equation can be written to depend on only one variable, we can use the separation of variable technique to solve this differential equation for the time for a particle to travel to a particular location x.
There are several advantages to the energy approach.
1) Energy is scalar while forces require vector math.
2) Energy conservation is a more fundamental physics law than Newton’s 2nd Law. Energy concepts are still valid in quantum and relativity.
3) You avoid integration when you want to find the speed of particle under the influence of only conservative forces.