Lecture Notes (Newton's Laws)

Transcription

1 Newton's First Law: Lecture Notes (Newton's Laws) - this law states that a body once set in motion and thereafter undisturbed, will continue in uniform motion forever - this law is a restatement of Galileo's Principle of Inertia - this idea was completely different than the prevailing views of the day, which were based on Aristotle's teachings - Aristotle believed that any motion, other than free-fall, of a heavy object required the exertion of a force - therefore, there can be no sustained motion without a constant applied force - this appears to agree with nature, but it is wrong, on this planet, gravity and friction obscure what is really going on - this idea was so difficult to grasp, that it took over eighteen centuries to get the true idea Newton's Second Law: - this law states that the acceleration of an object is directly proportional to the net force acting on it, and inversely proportional to its mass - this law will answer questions of what happens to an object that has a net force acting on it - for example, if you push a block of ice in a horizontal direction with a force of F, the ice will move with an acceleration a

2 - if you push this same block of ice twice as hard (2F), then the acceleration doubles, (2a) Newton's Third Law: - this law states that to every action there is always opposed equal reaction - forces are interactions between objects and always occur in pairs Types of Forces: - there are different types of forces, but all impart a push or pull on an object - some forces act on all objects, such as weight; weight is the force of gravity acting on an object due to its mass - some forces are associated with solids, such as friction, which is the force between two solids in contact with each other that resists their sliding across one another - some forces are associated with fluids, such as buoyancy, which is the force exerted on an object immersed in a fluid - the are also some apparent forces that are not truly forces at all, such as the "g-force"; this is not a force because it does not arise from an external object Force Basics: - an object that experiences a push or pull has a force exerted on it - the object is called the system; the world around the system that exerts forces on it is called the environment - there are two types of forces exerted by the environment on a system; contact and long-range forces

3 - contact forces act on objects only by touching it; Ex. a book on a table are each exerting contact forces on one another - long-range forces are exerted without contact Ex. a magnet can exert a long-range force without ever touching the object - each force has a specific cause called an agent; you must be able to identify the agent of each force; if you cannot name the force it doesn't exist - the symbol for force is F; the type of force is represented by a subscript letter; Ex. the normal force is F N Identifying Forces: - to help identify the forces acting on an object, we typically draw them; one way to draw them is by the pictorial method where we draw the object and it forces - another method of drawing forces is by replacing the system with a dot and drawing all forces in the direction of the force with their tails on the dot; this is called a free-body diagram - remember, when drawing your forces, you draw them as arrows; the length of the arrow is proportional to the size of the force; the direction of the arrow is the direction of the force Newton's Second Law: (cont.) - in order to study the motion of an object when forces are applied to it, we must perform experiments - experiments are easier to perform when the influences of friction and gravity can be avoided Ex. 1 in our constant acceleration lab, we used a cart on a track with very low friction Ex. 2 in our free-fall lab, we used a golf-ball with

4 little air resistance in order to calculate an accurate value of g; it would not have been good to use, say, a feather - in the diagram below, we seee a velocity-time graph of a rubber band pulling a cart; in this experiment, the rubber band was pulled at a constant force; as a result, the acceleration of the cart was constant - you can seee that the force was constant because the line on the graph is a straight line and not curved; also note thatt the acceleration was positive due to the positive slope of the line - if you stretch the rubber band out further, the force applied to the cart would be greater; as a result, the acceleration of the cart would increase - now take a look at the graph below; on this graph we see the slopes of three experimen nts of a rubber band pulling different numbers of carts

5 - as you can see, as you increase the number of carts pulled by the rubber band with the same force applied, the acceleration decreases - one can deduce thatt as the amount of mass increases, the acceleration decrease if force remains constant - we can generate the equation which relates force, mass, and acceleration: F = ma - the unit of force is the Newton (N) Combining Forces: - if two or more forces are acting on an object (system), we can represent these forces as vectors and draw them as such - another term, synonymous with the particle method of drawing forces, is the term free-body diagram - in a free-body diagram, you draw all of the forces as vectors, where all the tails of the forces end up on a dot (which represents the system) - because forces are vectors, the total force on the object vector sum of all forces exerted on the object is the - we can calculate the magnitude of thee combined forces by adding vectors, as we learned earlier; we call the vector sum of the forces on an object the net force

6 - the acceleration of an object is proportional to the net force exerted on the object and inversely proportional to the mass of the object being accelerated (F = ma) Problem Solving Strategy: (for Newton's 2 nd Law problems) - to find how the motion of an object depends on the forces exerted on the object, we must be able to first identify the forces on an object - draw a free-body diagram of the forces involved; you must show direction and size - add the force vectors to find the net force - use (F = ma) to solve for the acceleration - use the acceleration to plug into your constant acceleration equations to generate information on the velocity and position of an object Newton's First Law: - remember Galileo's principle of inertia; if an object is left alone, it continues to move with a constant velocity in a straight line (if it was moving), or it continues to stand still if it was at rest - Newton generalized this idea to come up with his first law of motion; it states that an object that is at rest will remain at rest or an object that is moving will continue to move in a straight line with constant speed, if and only if the net force acting on that object is zero - Newton's first law is often called the law of inertia - inertia is the tendency of an object to resist change; therefore, if an object is at rest, it tends to stay at rest; if an object is

7 moving at a constant velocity, then it will tend to continue moving at that velocity - equilibrium occurs if the net force of an object is zero; an object is in equilibrium if it is at rest or moving at constant velocity; remember being at rest is simply a constant velocity of zero - according to Newton's first law, a net force is something that disturbs a state of equilibrium; therefore, a net force changes the velocity of an object - as a result, a change in velocity (which is an acceleration) is the result of a net force acting upon an object Free-Body Diagrams: - because the net force on an object causes the acceleration of the object, we must know how to find the net force - the net force is the sum of all the forces on the object - we can find the sum of all the forces through vector addition of the forces drawn on the free-body diagram - free-body diagrams are diagrams used to show the relative magnitude and direction of all forces acting upon an object in a given situation - the size of the arrow in a free-body diagram is reflective of the magnitude of the force - the direction of the arrow reveals the direction which the force is acting - each force arrow in the diagram is labeled to indicate the exact type of force

8 - it is generally customary in a free-body diagram to represent the object by a box (or dot) and to draw the force arrow from the center of the box outward in the direction that the force is acting - an examplee of a free-body diagram is shown below - the free-body diagram above depicts four forces acting object upon the - objects do not necessarily always have four forces acting upon them; there will be cases in whichh the number of forces depicted by a free-body diagram will be one, two, or three - there is no hard and fast rule about thee number of forces which must be drawn in a free-body diagram; the only rule for drawing free-body diagrams is to depict alll the forces which exist for that object in the given situation - therefore, to construct free-body diagrams, it is extremely important to know the various types of forces; if given a description of a physical situation,, begin by using your understanding of the force types to identify which forces are present - then determine the direction in which each force is acting - finally, draw a box and add arrows for each existing force in the appropriate direction; label each force arrow according to its type

9 Ex. 1 A book is at rest on a table top. Diagram the forces acting on the book. Ex. 2 A girl is suspendedd motionless from a bar which hangs from the ceiling by two ropes. A free-body diagram for this situation looks like this: Ex. 3 An egg is free-falling from a nest in a tree. Neglect air resistance. A free-body diagram for this situation looks like this: Ex. 4 A rightward force is applied to a book in order to move it across a desk with a rightward acceleration. Consider frictional forces. Neglect air resistance. A free-body diagram for this situation looks like this:

10 Various Forces: Force Misconceptions: - some ideas about forces are misunder rstood because the dominated by friction; here are some common misconceptions about force: Earth is 1) When a ball has been thrown, the force of the hand that threw it remains on it. (False. Once the contact force is broken, the force is no longer exerted.) 2) A force is needed to keep an object moving. (False. If there is no net force, then the object keepss moving with unchanged velocity)

11 3) Inertia is a force. (False. Forces are exerted by agents in the environment, they are not properties of the object.) 4) Air does not exert a force. (False. Air exerts a large force, but it usually is balanced on all sides so it usually cancels out.) 5) The quantity ma is a force. (False. The equal sign in F = ma simply states that experiments have shown that the two sides of the equation are equal.) Applications of Newton's 2nd Law: - whereas Newton's first law predicts behavior of objects for which all existing forces are balanced, the second law of motion pertains to objects for which all forces are not balanced - Newton's second law states that the acceleration of an object is proportional to the net force applied to the object, and inversely proportional to the mass of an object - the algebraic representation of this is F = ma - Newton believed that this law was true for all motions; but during the early 1900's we have discovered that Newton's laws of motion break down in two areas 1) As velocities approach the speed of light 2) As the size of objects decrease to the atomic level Mass and Weight: - Galileo was the first scientist to correctly describe the nature of falling bodies; he said that no matter an object's weight, all objects gain speed at the same rate as they fall to the Earth - you can restate this fact by saying all objects have the same downward acceleration (9.8 m/s 2 )

12 - now let's calculate the weight force (F g ) exerted on an object with mass m - mass is a measure of resistance to acceleration (inertia); it is a scalar quantity associated with matter - the SI unit of mass is the kilogram (kg) - to calculate the weight force of an object we use Newton's second law F = ma - if you use a free-body diagram to show a falling object, you can describe the situation; disregard air resistance; there are no other forces acting upon the object, only the weight force F g - the object's acceleration is g, therefore, F g = mg - both the force and acceleration are downward; as a result, the magnitude of the object's weight is equal to its mass times the acceleration it would have had if it were falling - an object's weight varies depending on the magnitude of g; for example, g on the moon is only one/sixth that of the Earth; as a result, your weight on the moon is one/sixth your weight on the Earth - no matter what the value of g is, an object's mass is always the same Scales: - most scales contain springs; when you step on a scale, the scale exerts an upward force upon you

13 - - because you are not moving, the net force is zero; this results in an upward F sp equal to the downward F g the scale measures weight not masss Apparent Weight: weight is a force that is the productt of the mass of the object times the acceleration due to gravity (F g = mg) on or near the surface of the Earth, the acceleration due to gravity, g, is constant; therefore, the weight changes with a change in mass bathroom scales use a spring to determine the weight force; the spring provides an upwards force on your body that is equal to the weight force pulling you downward if you added forces to your person,, i.e. - hold a barbell, the scale would not give an accurate reading; the scale would read tooo high a weight force conversely, if you stood on the scale while holding yourself up on the bathroom counter, the scale would read a lower weight

14 - these two cases illustrate the concept of apparent weight; this is the force exerted by the scale - apparent weight may or may not be equal to your actual weight - the apparent weight of an object may even be zero; for example, if both the object and the scale were in free-fall, the scale would read zero - this is not to say that the object actually weighs zero, only that there is no spring force pushing up on the object - weightlessness means that your apparent weight is zero Terminal Velocity: - this is the constant velocity that is reached when the drag force equals the force of gravity - terminal velocity depends on the size, shape, and weight of the object Ex. a 200 kg cannon-ball will have a greater terminal velocity than a 0.5 kg golf ball Applications of Newton's 3rd Law: - in his first law, Newton described the behavior of objects when there are no forces acting on them or when the forces all balance, yielding a net force of zero - his second law explained how the motion of objects changes when the net force is not zero - Newton s third law added an original, new, and surprising insight into forces

15 - consider this problem: In a 100-m dash, an athlete goes to nearly top speed in less than 3 s from rest - we could measure the runner s mass before the dash, and we could use high-speed photography to obtain the acceleration; with the mass and acceleration known, we could find the net force acting on the sprinter during the initial acceleration - but where does this force come from; obviously it must have something to do with the runner himself; but is it possible for him to exert a force on himself as a whole - Newton s third law of motion, also called the law of action and reaction, helps to explain just such puzzling situations Newton's Third Law: - Newton's third law states: if one object exerts a force on another object, the second object at the same time exerts a force on the first object. These two forces, each acting on one of the two objects, are equal in magnitude and opposite in direction. - the startling idea in this statement is that forces always act in pairs, one force acting on one object, the other acting on anotherr object - a single force acting alone, without another force acting back on something else, does not exist in nature - for example, consider the sprinter; when the gun goes off to start the dash, his act of pushing with his feet back against the starting blocks (call it the action ) involves simultaneously a push by the starting blocks of an equal amount acting on him in the forward direction (call it the reaction )

16 - it is the reaction by the blocks that propels him forward - the action does not cause the reaction; the two forces coexist simply - if somehow the starting blocks came loose from the ground so that they cannot push back on his feet, they would just slide away when he tried to give them a big push, rather than providing the reacting force and the acceleration he needs to get started on the sprint

17 Third Law Misconce eptions: - a common mistake is to think that these action and reaction forces can balance each other to zero, and give equilibrium, as in the first law of motion - but in fact the two forces do not act on the same object; each acts on a different object, so they can tt balance out - it is like debt and credit; one is impossible without the other; they are equally large but of opposite sign, and they happen to two different accounts Examples of Newton's Third Law: - every day you see many examples of Newton s third law of motion at work; a car is set in motion by the push of the ground on the tires forward in reaction to the push of the tires on the ground backward - when friction is not sufficient, as when trying to start the car moving on ice, the car just spins its wheels in place because there is no reacting forward push of the ground on the tires - a tennis racket hits a tennis ball, accelerating the ball forward, even while the tennis balll exerts a force backward on the racket, causing tennis elbow in some cases,, or even broken arm bones

18 - the Earth exerts a force on an apple and the apple exerts an equally large force on the Earth; when the apple falls, pulled down by the gravitational pull of the Earth (the weight of thee apple), the Earth, in turn, is pulled upward by the equal but opposite attraction of the Earth to the apple - hence, during the apple s falll the Earth accelerates upward thoughh by only an infinitesimal amount - of course, we don t notice this motion of the Earth becausee of the difference in mass between the Earth and the apple, but the effect is there and in all similar situations - similarly, after the sprinter has left thee starting block behind and runs forward (owing to the force the ground exerts on his feet), the Earth moves a little in the opposite direction because of the force applied to it by his feet - on a small enough planet, this might become noticeablee Identifying Interacti ion Forces: - we can seee an example of interaction forces, below, where we have someone pressing a balloon on a door

19 - the interaction force between the hand and the balloon occurs when the hand comes into contactt with the balloon - we see another interaction pair between the balloon and the door - we have four interaction forces of: A) ) the force of the hand on the balloon, and B) the force of the balloon on the hand, C) the force of the balloon on the door, and D) the force of the door on the balloon - you can write these as F hand o simply as F A on B and F B o on balloon an on A nd F balloon n on hand or more - these two forces are known as action-reaction of thee hand on the balloon does pairs of forces - we must take note that the force not cause the ball to exert a force on the hand - these two forces exist together or not at all

20 Four Fundamental Forces: - up to this point, we have discussed mostly contact forces and one long range force (gravity) - these forces are part of four recognized fundamental forces - the four fundamental forces are: 1) Gravitational Interaction - all objects attract one another through gravitational interaction; this is an attractive force due to the masses of the objects 2) Electromagnetic Interaction - these include magnetic and electric forces; this is a force that holds together atoms and molecules; it is responsible for contact forces 3) Strong Force - this force occurs within the nucleus; acts between the protons and neutrons and holds the nucleus together 4) Weak Force - this force also occurs within the nucleus; responsible for radioactive decay - there is currently underway a major study to unify all four forces into one unified law; one of the major theories of this branch of physics is called string theory