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The Foundation of Newtonian Mechanics Abstract Due to the large number of equations involved in AP Physics 1, I (DCW) was curious to know what equations and key concepts were the absolute minimum starting information that the students need in order to have a chance to be successful in this subject. For Newtonian mechanics only, I went through the key concepts and derivations, and concluded that there is a very small set of starting information that the students build upon. This information consists of the four kinematic equations and Newton’s three laws. Because this information is so fundamental to student success, I recommend that students be pop quizzed frequently on this subject matter, to the extent that they memorize this information. Absolutely KEY definitions and concepts A proper understanding of Newtonian Mechanics absolutely requires a minimal amount of “working memory” (i.e., memorization) and understanding in the following areas: Definition of delta and the recognition of the appropriate symbol for delta Mathematical definition of average Definition of velocity Definition of acceleration A thorough understanding of the standard nomenclature A thorough understanding of the role of units and dimensional analysis A thorough understanding of the need for subscripts, and what those subscripts stand for A thorough understanding of algebra, separation of an unknown variable, etc. A thorough understanding of the four kinematic equations A thorough understanding of Newton’s laws CHAPTER 2 Displacement Displacement is defined as final position – initial position. In equation form, this becomes x x f xi (1) Velocity Velocity is defined as displacement per unit time, or x v t Acceleration Acceleration is defined as change in velocity per unit time, or v a t (2) (3) Kinematics 1st Kinematic Equation Equation (2) can be solved for Δx. x vt Since this displacement occurs for the whole time interval Δt, any changes in velocity that caused the resultant displacement are averaged out over the whole time interval. This leads to the following brief derivation: x vavg t (4) If acceleration for time interval Δt is constant, the average velocity in equation (4) is defined by its mathematical equivalent of 1 vavg ( vi v f ) (5) 2 A substitution of equation (5) into equation (4) leads to the first kinematic equation. 1 x (vi vf )t (6) 2 2nd Kinematic Equation Equation (3) can easily be solved for change in velocity, again assuming a constant acceleration: v a t (7) The delta-v term may be expanded, and equation (7) can be easily solved for final velocity when initial velocity, acceleration, and time interval are known. v f v i a t v f v i a t (8) 3rd Kinematic Equation Under some conditions, the final velocity is unknown, so equation (6) cannot be solved without additional work. Inspection of equation (8) indicates that it can be substituted into equation (6) to eliminate the need to know final velocity, resulting in the following: 1 x ( vi ( vi at )) t 2 Collection of like terms inside the parentheses yields: 1 x ( 2vi at ) t 2 Multiplying everything in parentheses by ½ yields: 1 1 x ( ( 2vi ) ( at )) t 2 2 Simplifying the above equation leads to the 3rd kinematic equation. 1 x vit a(t ) 2 2 (9) 4th Kinematic Equation Under some conditions, the time interval is unknown, so equation (8) cannot be solved without additional work. Inspection of equation (6) indicates that it can be solved for delta-t, and the result can be substituted into equation (8), resulting in the following: From equation (6) 2x ( vi vf )t 2x t vi v f (10) Equation (10) can now be substituted into equation (8) 2 x v f vi a v v f i 2x v f vi a v v f i 2 a x v f vi v v i f v f vi v f vi 2ax v 2f vi2 2ax v 2f vi2 2ax (11) Note that only the 1st and 2nd kinematic equations are mathematically independent. The other two kinematic equations must be derived via substitutions from the first two kinematic equations. For motion close to the earth’s surface, it is seen that vertical motion is accelerated while horizontal motion is constant. Due to this, the kinematic equations can be applied independently in the horizontal and vertical directions, leading to the following additional topics: One dimensional motion Projectile motion o The range equation (no change in elevation) o Changes in elevation o Path of a projectile Various projectile motion problem variable specifications are important for the fact that students must apply much of the algebra that they have previously learned in order to work those problems. CHAPTER 4 Newton’s Laws 1st Law Law of inertia – objects in motion stay in motion in a straight line until a net force acts on them. 2nd Law The acceleration of an object is equal to the net force on it divided by its mass. In equation form, (12) F ma Newton originally wrote this law differently. A slightly different derivation of Newton’s 2nd law, recognizing that acceleration is the change in velocity per unit of time, leads to the following: v F m t Expanding the delta-v term in the numerator leads to m( v f vi ) F t F mv f mvi t The product of mass and velocity is momentum, denoted by the symbol “p”. Since the final momentum minus the initial momentum is the change in momentum, p F (13) t Equation (13) leads to the following, which is the impulse-momentum theorem. Ft p (14) Note that momentum will be discussed and developed in following work. Also note that Newton’s 2nd law is seen in very many different contexts in physics. 3rd Law For every action, there is an equal and opposite reaction. Combinations Force x displacement The product of force and displacement, where the force is applied parallel to the displacement, is known as work. Work has units of Joules, and is an energy unit. CHAPTER 5 An object that is dropped from rest, accelerates under the influence of gravity, and gains velocity as it loses height. This means that gravity is applying a force to that object, and since the object’s displacement is parallel to the force of gravity, gravity is doing work on the object. This leads to the following: From equation (11) v 2f vi2 2 gy , where “g” is the acceleration due to gravity, delta-y indicates displacement in the vertical direction, and “down” is taken as the positive direction. Since the initial velocity is zero (the object was dropped from rest), v 2f 2 gy v 2f 2 g y Multiplication of both sides of the above equation by the object’s mass yields mv 2f mgy 2 1 mv 2f mgy 2 (15) The left hand side of equation (15) is known as kinetic energy (energy of motion), while the right hand side of equation (15) is known as gravitational potential energy. Equation (15) states that energy is conserved, and it says that as a falling object loses gravitational potential energy, it gains an equal amount of kinetic energy. The net work done on an object will accelerate it, based on Newton’s 2nd law. This leads to the following derivation: W Fx Substitution of mass times acceleration for F yields W max (16) Another substitution from equation (11) (the 4th kinematic equation) leads to the following: v 2f vi2 2ax v 2f vi2 2ax v 2f vi2 2 a x Substitution into equation (16) yields v 2f vi2 W m 2 mv 2f mvi2 W 2 W 1 1 mv 2f mvi2 2 2 W KE (17) Equation (17) is known as the work/kinetic-energy theorem. Force x velocity The product of force and velocity leads to the following: x Fx W Fv F P t t t (18) The product of force and displacement is work. Work per unit time is power. CHAPTER 6 Conservation of momentum Equation (14), the impulse momentum theorem, states that the product of force and the time interval over which that force was applied, causes a change in momentum for the object that the force was applied to. When two objects interact, Newton’s 3rd law states that the forces between them are equal in magnitude and opposite in direction. This leads to the conclusion that the change in momentum that one object causes to a second object is equal and opposite to the change in momentum that the first object experiences. Thus, due to Newton’s 2nd and 3rd laws, momentum is conserved. Conclusion 1 Based on the definitions of displacement, velocity and acceleration, kinematic equations 1 and 2 are derived. Substitutions from each of these equations into the other equation leads to kinematic equations 3 and 4. The combination of the four kinematic equations and Newton’s 3 laws leads to development of the following subject areas in Newtonian mechanics: One and two dimensional motion The impulse momentum theorem The definition of kinetic energy Conservation of energy Work and the work/KE theorem The definition of power Conservation of momentum These areas cover chapters 2-6 in the Holt Physics textbook (pre-AP physics, 203 pages of material). Thus, it is VITALLY important that physics students become intimately familiar with the kinematic equations and Newton’s laws. CHAPTER 7 Rotation There is a rotational equivalent of displacement, velocity, and acceleration. When these equivalent variables are substituted into the kinematic equations, the rotational kinematic equations result, as shown on the next page. Equivalent variables: x and θ; v and ω; a and ά Kinematics x v t a v t Rotational Kinematics t t (20) 1 (vi vf )t 2 1 x vit a(t ) 2 2 vf vi at f i t (23) vf 2 vi 2 2ax f 2 i 2 2 (25) x 1 (i f )t 2 1 it (t ) 2 2 (19) (21) (22) Memorization of the rotational kinematic equations is made MUCH easier if students are thoroughly familiar with the linear kinematic equations. In addition to the rotational kinematic equations, there are rotational equivalents for the variables associated with Newton’s laws: Torque is equivalent to force, and is equal to the product of force and lever-arm distance, where the force must be perpendicular to the lever arm. Moment of inertia, which is the rotational equivalent of mass, can be derived directly from Newton’s 2nd law F ma Fr mar a r Fr mr r Fr mr 2 I Thus, it is readily seen that moment of inertia, or I, is equal to mr2. A mathematical treatment of rotational variables in a manner exactly analogous to what was done above leads to development of rotational kinetic energy, rotational work, rotational power, and conservation of angular momentum. CHAPTER 8 Simple Harmonic Motion For a round object of radius R that is rotating at a constant angular velocity, one occasionally wants to know the position of a particular point on the object. Assuming that the horizontal axis is denoted as zero radians, it is obvious that the Cartesian coordinates of any given point are x R cos (26) y R sin (27) Since the object is rotating at a constant rate, ωi equals ωf, which equals ω. When ω is substituted into the rotational kinematic equation that describes this situation (e.g. eq (21)), the result is 1 ( ) t 2 t Since θi is equal to zero radians, Δθ is equal to θ. Also, the data for any related physics experiment normally starts at time zero, so Δt is equal to t. Finally, when the given rotating point gets to the vertical axis, it is as far from the horizontal axis as it is going to get. For the purpose of simple harmonic motion, this is known as its amplitude, denoted by the symbol “A”. Thus, substitutions into equations (26) and (27) lead to the following simple harmonic motion equations: x A cos t y A sin t Conclusion 2 Rotational relationships, while seemingly complex and abstract, can be developed in a manner that is functionally equivalent to the development of the linear equations outlined above. In effect, the rotational equations have an exact duplicate in the functional form of the linear equations, and the linear equations should be treated as a spring-board into the rotational equations. Thus, the linear kinematic equations and Newton’s laws in the linear world, are also vitally important for learning about rotation. Such a connection between linear and rotational equations leads to the following subject areas: Rotational motion Rotational kinetic energy Rotational work Rotational power Conservation of angular momentum Simple harmonic motion These areas cover chapters 7-8 in the Holt Physics textbook (73 pages of material). Thus, it is once again VITALLY important that physics students become intimately familiar with the kinematic equations and Newton’s laws, as these four equations and three laws are the key material behind almost 300 pages of the Holt Physics textbook. Physics Binder The physics binder should have several dividers in it, as shown immediately below. Standard nomenclature, with subscripts, superscripts, and units Definitions/Laws o Delta o Average o Displacement o Velocity o Acceleration o Vector resolution o Vector addition o Newton’s 3 laws o Rotational equivalent of linear variables Derivations and equations o 4 kinematic equations o Range equation o Definition of kinetic energy o Impulse/momentum theorem o Force x displacement and work o Work/KE theorem o Force x velocity and power o 4 rotational kinematic equations o Torque o Rotational kinetic energy, work, and power o Simple harmonic motion Key concepts o 1 D motion o Projectile motion o Conservation of energy o Conservation of momentum o Circular motion o Moment of inertia o Conservation of angular momentum o Relationship between circular motion and simple harmonic motion Standard physics models Kinematical Models Constant velocity Constant acceleration Simple harmonic oscillator Uniform circular motion Collision Causal Models Free particle: net force = 0 Constant force: net force = constant Linear binding force: net force = -kΔx “Center seeking force”, and force = constant Impulsive force