Physics 131
Physics Department
- Cal Poly
The Laws of Motion, Force, Energy,
and Momentum
Science is ultimately about how we know what we know.
References: Study Guide for University Physics
Physics 131 - SUPPLEMENTAL NOTES AND PROBLEMS by Ronald Brown, El Corral
The Supplement is about 50 pages covering kinematics, Newton's
laws, work and energy, momentum, and rotational motion. The notes are
intended to support the text - and focus on the fundamental concepts,
ideas, and principles related to each of the topics and on tying the
principles together and leaving you with a strong conceptual
understanding. Each section concludes with a set of conceptual
questions and problems to help your review.
Learning Center Hours: Mon. 9 am - noon; Tues., Wed. 9 am - 2 pm; Thurs. noon - pm; Fri. 9 am - noon.
NOTE: Hours may change when Center opens the second week of classes.
RESOURCES:
PHYSICS 131 NOTES:
Most of the ideas of this chapter you have seen before. Read over the chapter quickly, then re-read more carefully any material that is new to you. Don't avoid reading the historical note - it puts in perspective how our ideas of the solar system evolved.
Upon completion of the chapter (studying the material, doing the HW problems), you should be able to:
This chapter defines the basic properties, characteristics, and manipulations of vectors - quantities that have both magnitude and direction. Common vector quantities that will be used throughout your study of physics includes displacements, velocities, accelerations, forces, momenta, torques, angular momenta, electric and magnetic fields, etc. How to deal with vector notation, components, magnitudes and directions, and how to add, subtract, and multiply vectors is the subject of this mathematical chapter. It is important to being able to proceed to understanding the motions of objects in two and three dimensions.
Chapter 3 - ONE DIMENSIONAL KINEMATICS
The ideas of Ch 3 are central to understanding descriptions of
motion. It is essential to get both the connections and the
distinctions between velocity and acceleration. Knowing how to
manipulate the eqns for motion with constant acceleration will be
essential to your work in later chapters. Understand this material
well, and Ch 4 - motion in two and three dimensions - will be easy.
From your work in chapter 3 you should be able to:
Chapter 4 - INERTIA AND TWO DIMENSIONAL MOTION
This chapter takes the ideas of Ch 2 (vectors) and Ch 3 (one dimensional motion) and applies them to the description of motion in two and three dimensions. After making the general connections to displacement, velocity, and acceleration in two dimensions, the application is made to just two problems: projectile motion and circular motion.
Projectile motion should look very familiar! The problems will be very similar to the free-fall problems of chapter 2. That is, the only essential difference is that in addition to having a vertical component, there is also a horizontal component to the motion that must be considered. In MOST cases, the horizontal motion will have no acceleration associated with it - so the horizontal velocity is a constant. The vertical component of the motion is dealt with identically to free-fall problems.
Circular motion problems are interesting because they can involve acceleration even when the problem is a constant speed problem. That is, since the motion of an object moving in a circle is constantly changing its direction, the vector velocity is continually changing - which implies acceleration. The acceleration in this case always has a component toward the center of the circular path (and is hence called the "centripetal acceleration").
Upon completion of the chapter (studying the material, doing the HW problems), you should be able to:
Chapters 5 and 6 - PARTICLE DYNAMICS - Force, Motion, and
Newton's Laws
Chapters three and four dealt with the description of motion - kinematics - with emphasis on problems involving constant acceleration. What has NOT been stated, is what causes objects to accelerate. That is the essence of Newton's laws. That is, Newton's laws describe the principles that relate to why things move as they do. There are three laws that are important to understand - even to internalize as a part of how you think.
Note: You should read chapters 5 and 6 together - noting that ch 5 establishes the principles that will be used in ch 6 problems. Chapter 5 is about Newton's Laws - how the ideas fit together to explain how the motions of objects are affected by forces. Ch 6 then deals with how to set up and solve a myriad of problems involving forces and accelerations.
Newton's 1st Law: The Law of Inertia - Objects only change their motion as a result of the action of a net force (or unbalanced forces). Inertia is the property of an object which tends to resist changes in motion. A measure of the inertia of an object is its mass.In effect, the three laws tell you: (1) Objects only change their motion due to forces; (2) The change in their motion can be represented as an acceleration; and (3) The forces that cause accelerations are a result of the object interacting with other objects.Newton's 2nd Law: The Law of Motion - An object responds to a net force by accelerating in the direction of the net force. The acceleration that results is equal to the net force divided by the object's mass. Or stated in the more common form:
F=ma where F is the net force (and is a vector), m is the mass of the object, and a is the vector acceleration which is in the same direction as the net force. Most of the rest of the course will be ultimately connected to Newton's second law - either in terms of solving problems using F=ma directly, or by using the second law to develop the concepts of energy and momentum and using those concepts to solve problems. It is essential that you understand Newton's second law - and all of its subtlety - and not just know that F=ma.
Newton's 3rd Law: The Law of Interaction - All forces are interactions between two objects. That is, every force that acts on an object is caused by some other object - and that second object consequently has an equal and opposite force acting on it due to the first object. This important principle is probably the most subtle of the concepts in this course - and maybe the most important. The third law has to be understood to really understand how to solve Newton's second law problems.
Types of Force Problems
The types of forces encountered in these chapters include the gravitational force and contact forces. You can ALWAYS set the gravitational force equal to mg. (I would avoid using "w" for "weight" since the symbol "W" for "work" will be used in chapters 7 and 8.)To solve Newton's law problems, you must:Contact forces include compressional forces, tensions, normal forces and frictional forces. Note that compressional forces and tensions include spring forces (which can be either compressional or tension) as well as any "push" given to an object or a "pull" - as with a string or rope. Spring forces will be assumed to obey Hooke's law, where the force required to stretch or compress a spring is proportional to the amount the spring is stretched or compressed. That is, F=kx represents Hooke's law for springs, where k is a constant which represents the stiffness of the spring. Normal forces are compressional forces that are due to the interaction between two objects that are in contact. The word "normal" means perpendicular. That is, the normal force on an object is always perpendicular to the surface in contact with the other object. Friction is a contact force which is parallel to the surfaces that are in contact with each other. If the two surfaces do not slip, the force is called "static friction". If the two surfaces are moving with respect to each other (ie, some slippage occurs), the force is called "kinetic friction".
Circular motion problems ALWAYS involve an acceleration toward the center of the circle which equals v2/R. The acceleration is CAUSED by the vector sum of all the forces that act. That is, an object cannot be in circular motion UNLESS there is a force toward the center of the circle of magnitude mv2/R.
Identify all the forces that act on an object.
Draw a careful force diagram that shows all the forces as vectors acting on the object being described.
Establish a coordinate system (ie, x and y axes) that are parallel and perpendicular to any acceleration that might result from the action of the forces.
Resolve all forces into components along those x and y axes.
Set up a Newton's second law force equation for each component direction (and for each object that moves in the problem).
Then solve the equations in each direction for the resulting acceleration. In nearly all cases, the forces along one of the chosen axes will cancel - ie, there will not be an acceleration along one of the component axes - and there will be a net force along the other axis which can be set equal to the mass times the resulting acceleration.
The concept of energy is probably the most important in all of science. This chapter will introduce the concept by beginning with the idea of the work done on an object by the forces that act on that object - as a way to describe the process of transferring energy to the object. The most important idea of the chapter is the work-energy theorem. If you understand well how to define work (it is not just "force times distance") and understand that the work done by all forces acting on an object changes the kinetic energy of that object an equal amount, then you have mastered the essential ideas. The subtleties, including the ideas of conservative and non-conservative forces, potential energy, power, etc., are details that will help in problem solving and are covered in Chapter 8. But the fundamental idea of the chapter is that all of the work done by all of the forces can be accounted for in changes in the kinetic energy. The principle of the conservation of energy, which is important in all areas of science and engineering, can then be articulated in terms of the work-energy theorem.
This chapter further extends the ideas of Newton's laws to systems of particles by defining momentum as (mass)x(velocity) and then invoking both the second law for each part of a system and Newton's third law to see that forces INTERNAL to a system cannot change the momentum of the entire system (ie, internal forces change the relative motions of parts of the system, but not the total momentum). That means that in collisions (covered in Ch 10) between objects (the most common usage of these ideas) the momentum of the SYSTEM is conserved even if the individual momenta are not constant throughout the problem. This principle of Momentum Conservation applies even when the mechanical energy of the system is NOT conserved (for example in "inelastic" collisions).
Chapter 11 and 12 - ROTATIONAL MOTION
These last two chapters complete the basic ideas of classical mechanics. They will be a very useful REVIEW if you recognize that essentially every idea is simply an extension of things you already know into rotational motion rather than translations. Look for the similarities between the linear motion equations and rotational motion equations. Each idea or quantity in translation has a corresponding idea in rotation. (F=ma, for example, becomes torque=(moment of inertia)*(angular acceleration) or KE becomes KE of rotation, or momentum becomes angular momentum, etc.) The key to understanding the equations and how to solve problems is to recognize a simple relationship between arc length and angle: S=R*theta , ie, if the angle is measured in radians, the arc length is simply related to the angle (with R being the radius of the rotating object. That means that a similar relation exists between velocity and angular velocity and between acceleration and angular acceleration. After getting past some basic definitions and ideas, the ideas of torques, rotational kinetic energy, angular momentum, etc., can come very quickly.
This Week's Laboratory
[Look here each week for a brief description of the laboratory.]
This experiment explores how to determine the average value for a set of data and the standard deviation - and in particular how the standard deviation relates to the experimental uncertainty that occurs as a result of the random fluctuations in an experiment. The experiment itself is a trajectory experiment and involves the ideas of two dimensional motion. From this results, you shoudl be able to determine the relationship between the horizontal range of an projectile and the height from which it was launched horizontally.
Week 2: Exp. No. 3 - Equilibrium of Forces
This experiment will use a "force table" to set up a problem
in which three or four forces are acting at a point and are in
equilibrium. You should think of this experiment as an exercise in adding
vectors. The equilibrium situation is created only when the vector sum of
the forces is exactly zero. You will set up the experimental situation,
then show both graphically and analytically that the vector forces add to
zero.
Week 3: Exp. No. 7 - Acceleration on an Inclined Plane
This experiment will determine the acceleration of a low-friction cart
along an inclined track to show the relationship between the change in
velocity, acceleration, and the distance travelled. From your results,
you will be able to obtain a value for the acceleration due to gravity.
Week 4: Exp. No. 3 - Equilibrium of Forces
This experiment will use a "force table" to set up a problem
in which three or four forces are acting at a point and are in
equilibrium. You should think of this experiment as an exercise in adding
vectors. The equilibrium situation is created only when the vector sum of
the forces is exactly zero. You will set up the experimental situation,
then show both graphically and analytically that the vector forces add to
zero.
Week 5: Exp. No. 4 - Coefficient of Friction
The purpose of this experiment is to verify the proportional relationship between the frictional force and the normal force between a block and the surface it slides on. It will allow you to determine both the kinetic and static coefficients of friction using two different methods.
Week 6: Exp. No. 8 - Centripetal Force
In this experiment you will combine the ideas of centripetal acceleration
from the circular motion discussion and Newton's laws (both second and
third law) in order to verify that the force required to make something
move in a uniform circle is given by mv^2/R.
Week 7: Exp. No. 13 - Elastic Potential Energy
In this fun experiment you determine the work required to stretch the
rubber band on a "Rocket Ball" and then launch that ball vertically and
compare the elastic potential energy before launching with the
gravitational potential energy when it is at the top of its trajectory.
Week 8: Exp. No. 10 - Conservation of Mechanical Energy
This is a good experiment to test the principle of energy conservation. It is performed on the low-friction track and involves two masses - one on an incline and the other connected by a string over a pulley. The falling mass pulls the cart on the incline up the incline. When the work done by all forces is considered, we should be able to account for all of the mechanical energy of the system.
Week 9: Exp. No. 21 - Conservation of Momentum
This week's experiment involves both elastic and inelastic one-dimensional collisions. The experiment has you measure the initial velocity and the final velocities of two carts which are allowed to collide on a level low-friction track. Both elastic collisions and perfectly inelastic collisions will be done in order to test whether linear momentum is conserved in both types of collisions. You will also determine whether kinetic energy was also conserved in those collisions.
Week 10: Exp. No. 19 - Conservation of Momentum - the Ballistic Pendulum
In this experiment, a projectile will be fired into the swinging arm of an apparatus designed to demonstrate the conservation of momentum in perfectly inelastic collisions. It will also be determined that while total energy might be conserved both before and after the collision, during the inelastic collision itself, the energy of the system is not conserved even though momentum is.
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