These notes are five pages. A quick summary: Newton s Second Law, F = ma, is the only new equation for Chapter 4 and 5. A free body diagram is an essential step in organizing information to solve force problems. Circular motion problems are force problems, since an object in circular motion is accelerating. In 1687, Isaac Newton kicked the scientific enlightenment and subsequent age of discovery into high gear when he published Philosophae Naturalis Principia Mathematica. The book was very detailed and fairly complicated, but is still used today, more than 300 years later, as the foundation of the study of mechanics. In addition to an explanation of the new mathematics of calculus, Newton created the Laws of Motion, the Law of Universal Gravitation and he explained the details of circular (and elliptical) motion. We will use Newton s Laws of Motion in Chapters 4 and 5 (and beyond); circular motion, and its connection to Newton s Laws, will be covered in Chapter 5. We will save Universal Gravitation for later, in Chapter 11. Newton s Laws of Motion Newton created three simple laws through which the definitions of velocity, acceleration and force are connected. 1. An object moving with constant velocity will continue moving with constant velocity unless it is acted on by an unbalanced force. 2. An unbalanced force acting on an object will cause that object to accelerate (i.e. change velocity.) The acceleration is determined by the strength of the force and the mass of the object. F = ma 3. A force is an interaction between two objects, which apply equal and opposite forces to each other. Only the second law is quantitative, in that it defines how we measure force. The SI unit of force is kg-m/s 2 which is defined as a Newton and abbreviated N. One Newton is about a fifth of a pound, so a 200-pound person weighs about 1000 Newtons. Page 1 of 5
Special Forces Newton s Second Law, F = ma, is the only new equation in Chapters 4 and 5, and it is the only equation we will need to solve force problems. As with all problems, force problems will require that we organize our information (i.e. draw picture, label information, etc), then write appropriate equations, and then solve algebraically. To help us organize information in a force problem, it is useful to acknowledge that certain forces will appear in many problems; you will see them again and again. We can call these special forces, which is the label I give to any force that has a specific magnitude and/or direction. Chapter 4 Special Forces magnitude direction gravity mg down! normal n to the surface tension T along dir of string Our problems will include other forces that we will not classify as special. For these other forces we will use a generic label, such as F 1 or F 2, much as we used numeric subscripts for x v or t in Chapter 2 or 3. Chapter 5 adds one additional special force : friction. Friction is due to the interaction of two surfaces on a microscopic level. This is why a surface that seems smooth can still create a good deal of friction: microscopically that surface probably isn t so smooth. There are two types of frictional forces: Kinetic friction occurs when an object slides on a surface. The friction acts against the direction of the sliding, i.e. kinetic friction always acts to slow the motion of the object. Static friction occurs when an object would slide on the surface, but is held in place on the surface due to the friction. Static friction always acts opposite the direction the object would slide if the surface was frictionless. The magnitude of the friction force, in either case, is determined by two factors: the composition of the surfaces and how hard they are pressed together. The latter is simple: the normal force is a measure of how hard the surface pushes on the object, and therefore how hard the object pushes on the surface. So the normal force acting on the object is a measure of how hard they are pressed together. To quantify the composition of the surfaces, we define the coefficient of friction for he two surfaces. This is a unitless number, usually between zero and one. We use the Greek letter mu, which looks like µ, for the coefficient of friction. Page 2 of 5
And we can write the expression for friction as: force of friction = µ n Just as we use n for normal force and T for tension, we use a lower case f for friction force. With our two standard equations from our free body diagram, we will include f = µ n as a third equation. For any pair of surfaces, static friction will always be greater than kinetic friction. This essentially means it takes more effort to get something moving (i.e. overcome static friction to make an object slide) than it takes to keep it moving (i.e. overcome kinetic friction.) For this reason, the coefficient of friction is different for static and kinetic friction. Every pair of surfaces has a coefficient of kinetic friction, µ k, and a coefficient of static friction, µ s. The coefficient of static friction is the higher value of the two, because static friction is greater than kinetic friction for any pair of surfaces. We can then write force of kinetic friction: f k = µ k n force of static friction: f s µ s n Note that the force of static friction is not simply equal to µn. This is because the static friction force can vary and will always be only what is necessary to hold the object in place. That is, the static friction force will balance other forces that would otherwise cause the object to slide; the magnitude of the static friction force is always equal to the magnitude of the forces it must balance. Most problems that involve static friction will present some kind of limiting behavior, and will require that you consider the maximum possible static friction force. For example, the problem might ask you to find the maximum allowable speed an object can turn in a circle, or the maximum incline of a surface without the object sliding. These problems will require that you assume the maximum possible static friction force is being used to prevent the object from sliding on the surface. If so: maximum force of static friction: f s = µ s n Now we can add friction to our list of special forces : Page 3 of 5
Chapter 5 Special Forces magnitude direction gravity mg down! normal n to the surface tension T along dir of string kinetic friction f k opposite direction of slide static friction f s opposite direction object would slide Force Problems The rules for force problems will be similar to the rules we have created for previous problems. Of course you will have to draw a picture and label information. An important part of organizing your information in a force problem is to draw a free body diagram for every object in the problem. A free body diagram is also called a force diagram, because it is essentially a diagram of forces acting on an object. It is called a free body diagram because it is a diagram that is separate from your regular picture. You free the object from the picture just so you can analyze the forces acting on it. Rules for Drawing a Free Body Diagram After you have selected an object for a FBD: 1. Draw a rectangle, oriented the same way as the object in the picture. 2. Write the name of the object in the rectangle. 3. Label forces acting on the object by drawing arrows from the edge of the rectangle outward. Use the corners of the rectangle for forces in a diagonal direction. 4. Define the x and y axes for your FBD. One axis must be in the direction of the object s acceleration. 5. Split forces into x and y components where necessary; the original force is the hypotenuse of a right triangle. Draw the legs of each triangle along the defined x and y directions. 6. Write Newton s Second Law for the x direction and for the y direction. Note that acceleration will be zero in one of these directions. Page 4 of 5
The rules for drawing a FBD are a subset of the rules for force problems, which are: Rules for Force Problems in General 1. Draw a picture. Label information, including the direction of acceleration of all objects. 2. Choose an object or multiple objects for which you would like to analyze forces. 3. Draw a free body diagram for each object chosen. 4. After finishing each FBD (see rules above), you will have a list of equations. Count and solve. Circular Motion Problems from Chapter 5 follow the same rules as those from Chapter 4, but Chapter 5 problems include friction and/or circular motion. In Chapter 3, circular motion was introduced as one simple idea: an object moving in circular motion is accelerating, and the magnitude of that acceleration is a = v 2 / r We can now bring this idea into our study of forces. We know that forces cause objects to accelerate, so forces must be responsible for the acceleration due to circular motion. In other words, an object can only move in a circle if forces are causing it to do so. For this reason, circular motion problems are force problems. This is a simple idea, and it will hold true in Chapter 5 and beyond (i.e. later chapters.) Circular motion problems are always force problems. There are no new rules for circular motion problems, because the old rules (above, the rules for force problems) are all that is needed. When using the above rules for circular motion problems, be sure to include the following: When drawing your picture, always include the circle of the motion. Make sure you can see the radius of the circle, and be sure to label it (whether known or not.) I strongly recommend using r if the radius is unknown, R if it is a known value. When choosing the x and y axes on your FBD, one axis must be in the direction of acceleration. This means toward the center of the circle. When writing your equations, be sure to include a = v 2 / r Page 5 of 5