Lecture 4-1 orce, Mass, Newton's Laws Throughout this semester we have been talking about Classical Mechanics which studies motion of objects at every-day scale. Classical mechanics can be subdivided into several areas. The part of classical mechanics which studies classification and comparison of motion and tries to answer the question: How does the motion occur? is called kinematics. We have already quite succeeded in our study of kinematics. Now we are able to describe different types of motion including multi-dimensional motion and even rotation. ut we cannot study motion without understanding its reasons. This is why mechanics is not limited to kinematics only. The other part of classical mechanics which studies causes of motion is called dynamics. Dynamics is based on the three Newton s laws. General discussion of these laws is the subject of today s lecture. These are universal laws. ll macroscopic objects surrounding us in our every day life obey these laws. It does not matter if we are talking about motion of the Moon or about motion of an apple falling from the tree, the general principles of motion are the same. What, in your opinion, causes things to move? nswering the question what is the reason of motion, you might notice that every time when a body changes its velocity, it interacts with some other objects. or instance, if we are talking about a motion of the ball during any sport game, it changes its velocity, because it was pushed or cached by a player or because it was affected by gravitational field of the earth. The same occurs with King Kong falling from the skyscraper. He is affected by the earth's gravity. Of course, the nature of these interactions differs. In the case of the ball kicked by a football player, we have an example of the short-range interaction. It occurs only during a short period of time of the actual contact between the ball and the player s foot. In the case of King Kong falling from the Empire State uilding, we have an example of long range interaction, where King Kong is constantly affected by the earth s gravity. However, there is no actual contact between King Kong and the earth until he hits the ground. In this case we are talking about interactions by means of some field (gravitational field of the earth in this example). Exercise: Think about other examples of short and long-range interactions. Interaction with other objects causes physical bodies to change their velocity. This interaction is associated with force, which acts on a body. The relationship between force and acceleration is the key subject of Newtonian mechanics, named after Isaac Newton
(1642-1727), who first formulated this relationship qualitatively as well as quantitatively in the form of the Newton's laws of motion. Even before talking about Newtonian mechanics, I would like to emphasize once again the same aspect which we have already discussed during the first lecture. Newtonian mechanics provides essentially classical description of the surrounding world, which is only valid at certain scales. Even though it seems reasonable for us, based on our everyday experience, to think about everything as a perfect Newtonian machine, we should not extrapolate this picture on entire world. One can not use Newtonian mechanics at very small atomic scale, where quantum mechanics should be used, neither one can use it at the scale of universe, where General Theory of Relativity is applicable. lso it is not possible to use these laws for the motion at very high (comparable to speed of light) speed. In this case Special Theory of Relativity has to be used. ll other situations, known as everyday life, fit quite well in the framework of Newtonian mechanics. This is why, those laws are so important. 1. Newton's first law We already said that people use to extrapolate the "common sense" to situations where it does not work. or instance, it is a common sense that the moving body will eventually stop, if you do not apply some special force to keep it moving. Do you think that this is true? or many centuries people thought that the only natural state for the body is to be at rest, if no forces are acting on it. This misunderstanding was even supported by the teachings of ristotle, who put it in the basis of his natural philosophy. ut let us look at this issue closer. Suppose we have a hockey puck placed on the floor. If we push this puck across the floor, it will move for some time but eventually it will stop. Now if we do the same experiment not on the floor but on ice and if we push it in exact same way, it will move for much longer. s you can see, is not necessary to push the puck all the time during the motion for it to be moving. Moreover the same original push gives different resulting time and distance of motion at different conditions. If we somehow could improve those conditions making the puck and ice smoother, it will travel for even longer distance. In ideal case, if we could only remove the ice surface at all it will move forever. So the true reason why the puck is stopping is not because it was not pushed or pulled, but because it was affected by friction and air resistance. This means that for a body to move with
constant velocity is the same "natural state" as if it was not moving at all. This is the content of the Newton's first law, which states that body remains in a state of rest or constant velocity (zero acceleration) when it is not affected by other bodies. So, if a body is at rest it will stay at rest, if it is moving with constant velocity it will continue doing so until influenced by other objects. ctually there is no difference between the two situations of a body at rest or moving with constant velocity. Indeed, the question of velocity depends on the reference frame. The object may have completely different velocities in two different reference frames. or instance, if you are a passenger on the airplane and you are siting in your chair all the time during the flight, then your velocity is zero with respect to the plane, while it is definitely not zero with respect to the ground. On the other hand you can move through the plane during the flight. In this case your velocity will not be zero with respect to the plane, but it still will be much larger with respect to the ground. If this plane flies with constant speed and the distance of flight is not too large (so we can ignore the earth's curvature) and if all the windows are closed then there is no way for a person in the plane to know how fast this plane is moving or if it moves at all. The only time when you can feel that the plane is flying is when it changes altitude or experiences bumps. ut in such a case there are forces acting on it. This explains why some reference frames are called inertial reference frames. Simply speaking, these are the reference frames where Newton's first law is working. It is also often called the law of inertia. y inertia we mean a property of the body to keep its velocity constant until it is affected by an external force. 2. Newton's second law We have already mentioned word "force" several times, but we have never defined this term. Now it is the right time to do so. The force is the reason for the body to change its velocity or in other words to obtain acceleration. Think what are the main characteristics of force? It can be characterized in several ways. irst, since the force acts on a body, it is applied to this body and has a certain point of application. Second, it has the direction in which it is applied and, third, it has the magnitude which shows how strong it is. Physical quantities, having both magnitude and direction, are called vectors. ased on the properties of force, we can conclude that force is a vector. This means that forces must obey the same laws of addition as all other vectors. or instance, if there are several
forces acting on a same body, one can define the resultant force or the net force acting on the body as a vector sum of all acting forces: net (4.1.1) Returning back to the example about a puck on the floor, there are several forces acting on it. These include the pushing force, which we have applied in order to move it, the friction force from the floor or from the ice, the air resistance, the gravitational force, and the normal force of the floor. However, adjusting the force which we have applied to this puck, we can make it move with constant velocity, like no force is acting on it at all. This does not mean that there are no forces. That only means that the net force acting on the puck is zero. So, we can reformulate Newton's first law in a different way as: The body's velocity stays constant if the net force acting on the body is zero. Now let us notice that even if we apply a very same force it may affect different objects differently. Indeed we can push light hockey puck on the floor to make it move, but we can also push with the very same effort on a heavy box full of books and most probably it will not move at all. This means that the body's acceleration depends not only on the applied net force, but on some other property of the body as well. This other property is mass of the body. To see that, one can conduct an experiment by applying the same force to two bodies with different masses and measure the resulting acceleration of these bodies. However, if we wish to perform this experiment, we have to give the exact definitions of physical quantities, which we are going to measure, such as force. We know that to measure means to compare with standard, so we shall use something as a standard of force. or instance we can use a spring scale. Indeed the harder you push or pull on a spring the more compressed or stretched it will be. Compression or stretching of the spring is easy to measure. So, all you have to do is to calibrate your spring scale for measurements of force. s long as the force is not too large, there is a linear relationship between the change of the spring s length and force acting on the spring. If we now conduct an experiment, it will show that the resulting acceleration is proportional to the applied net force a (4.1.2) net and inverse proportional to the mass a 1. (4.1.3) m
rrows above the force and acceleration in equation 4.1.2 show, that they both are vectors having the same direction. In fact, this experiment is the only way to measure mass. It can be done by applying the same force to different objects and measuring their accelerations. This experiment shows that mass is the intrinsic characteristic of a body which comes to existence with the body itself. To measure mass we can perform the above experiment first applying the force to the mass standard and then applying the same force to the unknown mass. If one compares the resulting accelerations, he/she can conclude how many times the mass of the object is different compared to that of the standard of mass. Since, the direction of the resulting acceleration is the same as of the applied net force, the mass is a scalar. Moreover, mass is an additive quantity, which means that, if we apply the same force to two masses combined together, we will still have equation 4.1.3 valid with mass as the sum of the two masses. Now we can formulate Newton's second law: The net force acting on the body is equal to the product of the body's mass and the acceleration produced by this force or net ma. (4.1.4) pplying this equation one has to remember that we are talking about the net force acting on the body not just about any force. We also have to remember that this net force acts on that body, not just any body in the system. Equation 4.1.4 also tells us that if the net force acting on the body is zero, then the resulting acceleration is also going to be zero, which brings us back to the statement of the Newton's first law. In this case, we say that the body is in equilibrium and all the forces acting on it are in balance. In some particular reference frame we can distinguish between dynamic equilibrium, when the body moves with constant speed, and static equilibrium, when body is at rest. Equation 4.1.4 also helps us to define the SI unit of force, which is 1 Newton (N )= 1kg 1m s 2 the force providing acceleration of one meter per second squared acting on the mass of one kilogram. There are other units of force in different systems of units. or instance, ritish unit of force is Pound (it is not unit of mass as people say sometimes). Talking about forces in nature we can distinguish between several main types of forces or interactions. Those include: Gravitational interaction, Electromagnetic
interaction, Strong nuclear interaction and Weak nuclear interaction. The greatest task of physics is to unify all these interactions under the same basic law. Since Newton's second law involves the net force acting on the body, in order to solve any problem using this law, first of all one has to draw a free-body diagram. This diagram shall include all the forces acting on the body, but it shall not include any forces acting on other bodies. It also shall include the net force, which comes as a result of the vector summation of all the forces acting on the body. One also needs to include a coordinate system, since the most effective way to add vectors (in this case forces) is by adding their components. So, the difficulty level of the problem very much depends on the correct choice of coordinate system. The Newton's second law may be applied not just to one body but to a system of bodies considered as a whole. In this case you will only need to include external forces acting on this system. ut you should not include any internal force acting between the different parts of this system, since they cannot provide acceleration to the entire system. or instance, if you are considering a train pushed by a locomotive as the whole system, you should take into account the pushing force of the locomotive as an external force. ut you should not include the internal pulling forces between the different carts in the train, since you are considering them as all moving together. pplying equation 4.1.4 one has to remember that this equation is a vector equation, so it can be presented in terms of vector components as net, x net, y net, z ma, x ma, y ma. z (4.1.5) So, one can see that the component of acceleration along the certain coordinate axis is only caused by the net force s component along the same axis. 3. Newton's third law We just have considered several examples of interacting bodies, such as the floor and the box on the floor or the train consisting of several carts. We also said that one should not include forces acting in between the carts to the Newton's second law equation when considering the train as a whole, since these are internal forces. Now let us take a closer look at these forces acting between the different objects. If we have two carts in the train, let us say and, and the train moves from the left to the right, having cart
before cart, then cart pulls on cart with force directed from the left to the right. t the same time cart pulls on cart with force directed from the right to the left. The similar situation took place in the previous examples, since the box was pulling on the floor with its weight, while the floor was pulling on the box in the opposite direction with normal force. Newton's third law has to do with these pairs of forces and it states that When bodies interact, the forces on the bodies from each other are always equal in magnitude in opposite in direction or. (4.1.6) Note that the statement of the third law is not trivial and cannot be reduced to the special case of the second law. This is because the second law tells us about forces acting on the same object, while Newton's third law tells us about the forces applied to different objects. orce acts on object, while force acts on object and also called the reaction force to. Let us continue with this example about the train. We shall call the force of the locomotive acting on the first car from left to right to be and we will neglect friction and air resistance, so the Newton's second law for the first cart will be m a, m a, where we have chosen the direction from the left to the right as positive direction. The Newton's second law for the second cart is m a, m a. We can add these two equations, which gives m a m a ( m m ) a, a m m., In this equation we have used Newton's third law, which allowed us to cancel forces and, so we ended up with the Newton's second law for the entire train and found its acceleration. If the third law was not true, one could not apply the second law to each of the carts and to the entire train at the same time.